20 rounds? Every two minutes the slowest get's knocked out!!!
Single-driver elimination didn't work in the two races it was used in 2016 because people just did a run at the very beginning, and then only bothered going out again when they were about to be knocked out. At that, they would only if they thought they could substantially improve on their time. This meant they tended to do 3-9 laps - less running than in the 1996-2002 12-lap format, let alone the current three-session eliminator method (which requires at least 9 laps but more often has the top runners doing 15).
4 sessions is not so bad but I think at least one session would turn into a formality most races. This sometimes happens now (witness Q1 in China, where you just had to be the last non-Williams to set a time to go through, removing most of the drama). I'm not sure if there would be any true jeopordy for the top teams unless one session had 7 or fewer entrants (and in that case, we'd probably see more strategic non-completions of Q3 because of the big advantage of a fresh set of tyres and the reduced advantage of being in the next session).
If the 4-session eliminator happened, I assume that to avoid perverse incentives, only those doing Q4 have to start on old tyres (namely, the ones on which they set their Q3 time).
Fortunately, there is apparently only one team interested in the system, which means it is unlikely to get the necessary approvals to be adopted. Expect the current system to stay.