It is not a known fact but it follows with logic that this was his fate. In other words, it is very plausible. Navalny was a fierce Putin critic. Why did he even have to go to prison?

It is assumed. It kind of goes with the territory. It was reported that he was feeling ill & collapsed. "Attempts" to resuscitate him failed. Brutal conditions in the gulag can lead to these kinds of sad endings for freedom fighters in Russia. What can anyone believe coming out of Russia...or RT.

But for Putin, he would still be alive.

Either directly (Putin might have been feeling pretty frisky after the Carlson interview) or indirectly (given conditions in the Siberian gulags) Putin was responsible.

Polar Wolf Prison is not a place you come out of alive.
It is Putin's storage facility for undesirables and was the latest in a series of increasingly worse detention facilities Alexei Navalny had been sent to since his arrest.
It dates from the soviet era when it was known as detention area 5.
Even in summer the temperature never reaches above minus 25 degrees centigrade, the food is basic rations, and those sent there never survive their full sentence, most expiring of frost bite, exposure or starvation, yet escaping means death, you will freeze to death after only an hour outside of the facility.
It is a death camp.

It's a reasonable question. The answer is that Navalny was illegally detained on Putin's orders. So anything that happened to him was the responsibility of Putin. Period.