Normally when that T-CON board T400HW01 V3 / 40T02-C05 / 55.40T02.C06 goes bad the result is color distortion, "solarized" images or , in some cases, washed out picture, but in this case customer reported that the TV would start normally, play sound, but nothing would show on the screen and then it would shut down on its own.
We received the T-CON board , stuck it in our TV and got the exact same result - the TV would come up for a few seconds, no image at all, then it would shut off.
Turned out the board was pulling too much out of the 12V power supply rail due to a problem.
When we located and resolved the reason for that and put the board back in the TV it still did the same thing.
It turned out it was not supplying 12V to the T-CON board anymore, so we pulled out the main board from where the 12V come and started tracing.
Eventually, it turned out that there is a fuse on the 12V rail to the T-CON T400HW01 V3 / 40T02-C05 / 55.40T02.C06 on the main board and that it was blown open.
It's a 5A / 24V pico fuse that we did not have handy, so we stuck a different one:
This is the back of the main board from a Sony KDL-40Z4100. You can see the label of the 12V fuse F1422 and our patch over it.
After installing a replacement fuse as shown above the TV stayed on and we got a good image with our own testing T-CON board .
Alas, customer's board was still showing image with too much brightness / wash out, which was not caused by the usual problem in the alpha control IC; we are still trying to figure that out and will eventually succeed.
But you should know that if your Sony KDL-40Z4100 tends to shut off within a few seconds after it is turned on and shows nothing on the screen even though the backlight comes up then you should check that fuse on the main board.