Can an MP who didn't vote for the 2nd reading of a bill propose amendments to it?

Question

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.)

Answer 1

It's not true. The rule would be completely unworkable; Opposition MPs would be unable to table any amendments to government bills, since they'll almost inevitably have voted against the second reading of the bill. All rules I've found regarding tabling of amendments simply refer to "members", making no other distinction. They might control the timings or method of tabling amendments but none that I can see actually segregate specific members.

An example to hopefully solidify that. Here's the Second Reading division of the European Union (Notice of Withdrawal) Bill. Note the presence of David Lammy in the Noes list (I.E. he voted against second reading). Here's the list of amendments tabled to that bill, and note Lammy is one of the sponsors of a tabled amendment. Ditto Alasdair McDonnel aaaaand a bunch of others.

Edit:

I guess the "propose" wording might limit the point to the 6 members listed at the top of amendments. Note Helen Hayes is one of those 6 for one of the tabled amendments and also voted No in the division.