Bean–Tannock–Ulfarsson maximal staircase-encoding conjecture

Let $J(\ell,k)$ denote the number of $132$-avoiding permutations of length $\ell$ whose staircase encoding has $k$ nonzero boxes, and let $C_i$ denote the $i^{\text{th}}$ Catalan number. For fixed $\ell$, let $k$ be the largest value such that $J(\ell,k)\ne 0$. Bean–Tannock–Ulfarsson's conjecture. If $\ell=3i+2$, then

$$J(\ell,k)=C_i,$$

and if $\ell=3i+1$, then

$$J(\ell,k)=\frac{3}{2}{2i\choose i}.$$

The paper states that this conjecture is proved using the preceding generating-function theorem and the formula for $J(\ell,k)$, so the conjecture is resolved.