Thank you!
The CPU is actually an Ivy Bridge.
Oops, I was a little sleepy when I answered I guess!
Its an Intel Core i7-3517U. 
Well, it’s not that, BIOS 204 even supports that CPU, so it’s not a microcode issue. I will make you some BIOS to try with updated ME FW, maybe that will sort the issue out, hopefully nothing will be carried over from a broken ME source.
Thank you both, @Lost_N_BIOS and @LittleSnail.
I tried the firmware attached by LittleSnail, but both the Windows program and flashrom fail to verify it once flashed.
I’m having a few problems with my test clip (it was really low quality and is not able anymore to make proper contact with the 8 EEPROM soldered points): I ordered a new one and should receive it in a few days. I’ll then be able to experiment a little more with other .bin files.
Hi there,
first of all happy new year!
I just got the new test clip today, so I can get back experimenting.
Any new idea on what I could try?
Thanks!
I just tried: this time it is verifying correctly, but the computer is still showing no sign of life after having flashed the .bin file that @LittleSnail provided.
@Caigo - First, lets manually confirm your writes. And you have to redo it, since you tried to start system it may be changed now, or you can do it with the laboneinside BIOS
Erase chip, blank check, then open BIOS and write to chip. Once done do not try to start system. Close application. Open application, read chip, verify buffer, save file.
Open that file and the one you wrote to chip in hex editor compare function, if they match write is OK
If they do not match then you found the issue (failed write + false verification) <<This happens if wrong software version for the chip, or wrong ID used etc (For W25Q64FV be sure you are using 1.30 or 1.34 and W25Q64BV ID
If they match, and laboneinside BIOS does not start the system you may have other issues, something shorted previously (or currently, in place live short) or something blown out at previous short, may have fried something when the power went out on you (ie surge through the line etc)
If no reaction at all to power on button attempts, then this may very likely be the issue
@Lost_N_BIOS - I tried to do as you suggested (erase, blank check, BIOS write; close and re-open application; read chip, verify, save). I did it all with 1.34, selecting W25Q64BV ID.
Unfortunately, checking the files (flashed with read and saved) with HxD (https://mh-nexus.de/en/hxd/) I saw they are the same. I tried both with the file provided by @LittleSnail and with the laboneinside one. Same result.
I guess I should start searching for a new computer then, no?
Thank you so much for your help, it was really much appreciated.
@Caigo - Yes, it does sound like it’s not the BIOS that is the issue then, sorry to say 
Maybe you can find board on ebay.
* edit - Yes, I see board with w/ i5-3317U is only $68, I would go this route instead of a whole new system, it will be much cheaper.
@Lost_N_BIOS - Uh, that’s a pity. But thanks for the suggestion!
Unfortunately I’m based in Europe, and the cheapest UX31A motherboards I can find on Ebay are running at 105-110 euros (around 120 US dollars). Still cheaper than a new machine, though some refurbished newish X-series Thinkpad (e.g. x250 with fHD screen, i5, 256 GB SSD) would run at just 250-300 euros.
I’ll consider what to do.
One more option would be getting a third-party unofficial motherboard for the Asus from China on Aliexpress (cheapest ones are around 75-90 US dollars), though I’ve had no experience with such parts and have no idea on their actual quality.
@Caigo - Were you on ebay.com? Both listings I looked at, and there was many others around same price, anyway both have shipping worldwide that I clicked into (one does not ship to France)
https://www.ebay.com/itm/151450117760
https://www.ebay.com/itm/254376327896
What does third party unofficial mean? Is that some OEM non-Asus board, or just an Asus board being resold by someone (same as ebay)?
@Lost_N_BIOS - I checked both ebay.com, ebay.de and ebay.it (I’m ordering from Italy). The thing with imported boards is that the shipping expenses are of $50. If I don’t find anything else I may pay them, even if it’s a little weird to pay almost as much as the actual price of the component just for the shipment. 
I found some cheaper ones ($70-90) on Aliexpress, but I think they’re OEM non-Asus boards (I meant this, sorry for my poor phrasing).
Here are two of them for instance:
https://aliexpress.com/item/32862050120.html
https://aliexpress.com/item/32904835118.html
The boards on ali dont usually have any stickers on them,or theyre blacked out with marker pen,so theres no serial,model number,or for some boards the mac address.That can be a problem sometimes as the same board,with a different revision could have the bios for a different model.The bios usually don`t have a serial number or a windows key (win 8) . If you buy one and it works, i would sell the laptop and buy something else.
@LittleSnail - Thank you for your feedback!
At the end I actually decided to get a second hand laptop instead: the uncertainties in buying one of those boards from AliExpress were maybe a little too high (quality of the item, and really long expedition times mainly). I’m currently scanning Ebay for T- and X-series Thinkpads waiting for a good deal.
In the future I may try to recover the Asus as a secondary laptop, or just scavenge it for parts (possibly the display, the keyboard if I find a controller; the SSD has an uncommon standard but may be recoverable thorugh an adapter, and the RAM unluckily is soldered).
Thank you really much for your help!
Keep scanning ebay for a spares repair UX31a laptop,or used motherboard.
Yes, I’m saving the research.