The grid correspondence conjecture for permutation patterns and fork posets

Let $J_s$ be the permutation-pattern matrix obtained from the identity matrix by moving its last column to the beginning, and let $\bigvee_s$ be the poset with elements $a,b_1,b_2,\twoheadrightarrow,b_s$ satisfying $a<b_i$ for every $i$. Write $ex(k,l,M)$ and $sat(k,l,M)$ for the extremal and saturation numbers of a forbidden $0$-$1$ matrix $M$, and write $La^*([n]^2,P)$ and $sat^*([n]^2,P)$ for the strong extremal and saturation numbers of a poset $P$ in the grid. The grid correspondence conjecture. For every $se=3$,

$$sat(n,n,J_s)=ex(n,n,J_s)=sat^*([n]^2,\bigvee_s)=La^*([n]^2,\bigvee_s).$$

A copy of $J_s$ corresponds to an embedding of the fork poset, motivating the expected equality between the matrix and strong grid quantities; the first part was already stated in the cited literature, while the full correspondence remains conjectural.