Unbounded rank gaps in differential posets

Let $P$ be a differential poset, and let $p_n$ denote the number of elements in rank $n$. Define the rank gap by $\Delta p_n=p_{n+1}-p_n$.

Unbounded rank-gap conjecture. In any differential poset,

$$\lim_{n \to \infty} \Delta p_n = \infty.$$

The conjecture asks whether rank sizes in every differential poset eventually have arbitrarily large successive differences. The surrounding discussion gives partial lower bounds for particular differential-poset constructions, but does not establish the general assertion.