[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.