Will a transaction with no priority fees use up the gas limit?

From ledger's website:

You can send transactions without adding a priority fee, however, keep in mind that setting a very low priority fee is risky—transactions may wait in a pool until the entire gas limit is used up.

This confuses me. I (think I) understand the concept of gas limit, and I dont see how it applies here. My understanding was that, until a transaction is picked, it sits in the mempool and doesnt burn any gas. If it's picked, then only gas limit matters, not Max Fee nor Max Priority Fee.

In other words, is there a risk to setting max fees that are too low apart from getting a stuck transactions?