Why Is the Constant Sum Model Susceptible to Being Fully Drained When the Price Peg Fails?

The constant sum model ($x+y=k$) maintains a 1:1 price ratio regardless of the actual market price. If one token's market price falls below the peg (e.g.

$1 token A neq 1 token B$), arbitrageurs can continuously deposit the cheaper token and withdraw the more expensive token at the pool's guaranteed 1:1 price. Since the pool's price never adjusts, the arbitrage is infinite until the pool is completely drained of the more valuable token.