will off chain ETH tokens be impacted by the merge?

I have a fair amount of W.ETH on the AVAX chain. Is there any reason to think that bridges tokens' value may be effected by the merge?

I'm worried about the PoS ETH and PoW ETH being able to cross the bridges and doubling the ETH supply. Is this something I should be concerned about? Should I move all ETH tokens to the ETH chain to make sure I don't lose any value?