Storj Labs still pays for the transactions, but the cost is much lower, because these are L2 transactions. There was some discussion about the withdrawal costs being higher for SNOs. But from what I can tell the difference is negligible in practice.
I looked back at my previous transfers to binance for example and L1 transfer to that address cost 51496 gas. A withdrawal from zkSync to that same address costs 52700 gas. So practically no difference. And the upside is that zkSync will allow you to pay those fees in STORJ. Which saves you the effort and transaction fees of getting ETH into the wallet to be able to move the STORJ around.