The asymptotic forbidden fork-pair conjecture for induced subposets

Let $Y_{k,r}$ denote the $r$-fork with a $k$-shaft poset and let $Y'_{k,r}$ denote its dual. Let ${\rm La}^{\sharp}(n, \{Y_{k,r},Y'_{k,r}\})$ be the maximum size of a family of subsets of $[n]$ containing neither $Y_{k,r}$ nor $Y'_{k,r}$ as an induced subposet, and let $\Sigma(n,k)$ denote the corresponding extremal quantity defined in the paper.

Forbidden fork-pair conjecture. For all $k \ge 2$ and $r \ge 2$, there is an $n_0=n_0(k,r)$ such that if $n\ge n_0$, then

$${\rm La}^{\sharp}(n, \{Y_{k,r}, Y'_{k,r}\}) = \Sigma(n,k).$$

This proposes a generalization of the theorem proved earlier in the paper for the case of the ordinary fork. The parser supplies no evidence of resolution; the conjecture is therefore recorded as open.