I never met any person who has not claimed in a professional set to be swamped with work. It's unwise to seem not busy. There is no signal here, just socially acceptable noise.
It will likely never improve
The point is that they have no real understanding of time it takes to develop anything. It all comes from what the developers tell them. You have a huge asymmetric information advantage and should be able to use that to limit the amount of work you have to do.
To put it bluntly, if you work 45 hours a week now and want to work 40 start padding you estimates by 10%. Nobody is going to know.
That would be a wonderful strategy if they asked for estimates.
Even if engineering has literally 0 planning input at all they're still the ones doing the work. I've never seen a PM sit behind an engineer 45 hours a week and make sure their working. Just stop working and go home or take a longer lunch if you want to work less. They're not going to fire you. The logic doesn't make sense. Firing an engineer on a project behind schedule is going to make the problem worse. Not better.
If you are legitimately worried about being fired you're in the top 1% worst organizations and it's time to move on.
[1] It would not be an exaggeration to say that this division has literally never produced value for the company.
> There is no noise here, just socially acceptable noise.