I can confirm that it works
We took the bus yesterday at 10 Pm from the west bus station in Bishkek. Actually the bus departs just behind the big building next to the busy west bus station, where all the mashrutkas depart.
There was another, earlier bus too, I think around 6.30 pm. We payed 2000 som for the tickets (we bought them one day in advance though). The bus driver takes another 100 som for every piece of luggage that you want to put in the trunk.
After two hours, you reach the Kazakh border, where it is a bit chaotic. You step out of the bus an pass the Kyrgyz and Kazakh customs. It took us two hours to do so.
Then you have a couple of hours, where you can try to sleep. For me it worked quite well. In the morning the bus reaches the Uzbek-Kazakh border. We had to spend another two hours there.
After 13 hours the bus reached the taschkent bus station.
So, I hope this helps some of those who plan to do the same trip!