It's been said in a (now-deleted) comment here that
You are not allowed to propose an amendment to a bill unless you voted for the second reading. This means there is a strong incentive to vote for the second reading, which may well not carry over to the final vote.
I tried to verify this from external sources (e.g.) but insofar I could not. So is this true, and if so what specific precedents or explicit rules (e.g. Standing Orders or Erskine May paragraphs) is it based on?
(Note that this is not a good question for Skeptics because the claim is not "notable" enough by their standards, being made in a comment here.)