How do you think the ceasefire suggestion will be received ?
However Putin wants it to be received.
The main satisfaction has already been achieved - Ukraine was seen to have humiliated itself, and its western partners.
So if it's only 20% of Ukrainian territory that will be absorbed, instead of 80% or 100% makes little difference.
The people, though largely stupid, still understand this was a titanic effort for a country with Russia's currently limited capabilities to achieve so much. They understand - if only instinctively - that some clever diplomacy and shadow play was involved in extracting this - and they will ascribe this all to Putin himself.
The simple math of Russian power vs Western power could not have produced this. The people understand the Kremlin, through its 1,000 year experience has pulled off something we could not even imagine.
This is why we trust it. Betting against the Kremlin rarely proves worthwhile, even for internal doubters. Those guys can pull a rabbit out of a hat in the unlikeliest of circumstances.