The height-two poset linear-extension almost-completeness conjecture

Let $e(P)$ be the number of linear extensions of a finite poset $P$, and let $e'$ denote its restriction to posets of height two. Write $\mathcal T_{e'}$ for the set of values attained by $e'$. Height-two linear-extension conjecture. The function $e'$ is almost complete, meaning that its value set contains all but finitely many positive integers. Numerical evidence exhibits many missing values, but height-two posets have sufficiently many possible inputs and the restricted function can attain large primes, so the conjecture remains plausible and open.