Is your cost basis wiped/lost if you transfer from an exchange to cold storage then back to an exchange?

Basically the title.

Reason for my question is that a couple years ago I sold about $1,000 worth of BTC that was purchased and stored on Coinbase. Because Coinbase had my transaction history and cost basis, they automatically sold my most expensive BTC so I was able to write off a (modest) realized loss from my taxes in a tax-loss harvesting scenario.

Since then, I've transferred all of my BTC to cold storage. I have no plans to sell any time soon because I've grown to realize that the best thing to do with BTC is to simply always be buying and HODL.

But it got me wondering: If I transferred my BTC (or a portion of it) back to Coinbase, is my cost basis wiped away? I would assume so, but wanted to confirm.

Then a follow up question: If I did transfer back to Coinbase and sold a portion, what would be used as the cost basis when calculating how much gain/loss I would taxed on?

Thanks in advance!