Stanley's strict rank growth conjecture for differential posets

Let $r$ be a positive integer, let $P$ be an $r$-differential poset, let $P_n$ denote its $n$th rank, and write $p_n:=|P_n|$ for the rank cardinalities. Stanley's conjecture. An $r$-differential poset $P$ has

$$p_1<p_2<p_3<\cdots.$$

The conjecture asks whether the weak rank growth known for differential posets is always strict. The paper establishes this strict growth, so the conjecture is solved.