[User3456335]

The paper seems to have a definition where bed posts are never deleted, i.e. they are all assigned probability 1 in which case the conjecture is obviously true.

The counterexample seems to rely on correlations between edge deletions which makes no sense because the deletions should be independent (in the definition I'm seeing on Wikipedia).

I could be wrong here because I haven't read it in detail but on first sight, it looks like there are some serious issues with mathematical rigour here.

[JohnKemeny]

You can probably replace any transversal vertex with a large clique (depending only on p) such that the probability that the clique remains connected and at least one post remains.

(I.e., you can "model" undeletability by adding sufficiently many twin posts.)

Ps., do you mean "in which case the conjecture is obviously false"?

[User3456335]

Surely, if the poles can't be deleted then which node is chosen from a bunkbed will not affect the connectedness probability, which would make it impossible to find a counterexample, right? What am I missing here?

PS Depends on what conjecture we refer to here.