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"?
(I.e., you can "model" undeletability by adding sufficiently many twin posts.)
Ps., do you mean "in which case the conjecture is obviously false"?