Are centralised exchanges staking users crypto but not passing on the rewards?

Some exchanges seem to support the purchase, sale and transfer of many PoS cryptos.. but as yet don't appear to support staking either across the world or regionally.

I'm wondering if they're most likely genuinely still working on adding support or simply it's a convenient excuse as they are staking the crypto in the background and collecting the rewards for themselves? Also wondering if there's any way to prove or disprove this.

You'd have thought if any exchange was adding support for PoS crypto, that would also have to include support for staking at the same time.

Some are taking a very long time, so it does make you wonder.. a bit like what happens to all those unsupported airdropped tokens...

Edit for clarity ????

Please note not really questioning the fact exchanges take a big cut of rewards as some comments correctly point out.. rather are they taking 100% of rewards by saying they don't yet support staking but in fact stealthily staking in the background with customers tokens.

For example the likes of Coinbase listed Cardano on March 21

https://blog.coinbase.com/cardano-ada-is-now-available-on-coinbase-dd30c1e0d93a

However they never announced staking support until a year later...

https://financefeeds.com/cardano-staking-rewards-go-live-coinbase-exchange/

Did they in fact stake for that year gap and keep 100% of the rewards for themselves and if so any way to prove/ disprove?

Again currently Coinbase also support Solana.. yet they don't support staking.. or do they and they're just not paying out.....