Skew shape conjecture for diagonal pattern avoidance

Let $S$ be a skew shape, and let $k\geq 1$. For a filling of $S$, write $\operatorname{Tr}(S,\pi)$ for the number of transversal fillings avoiding the filling pattern $\pi$, and let $\iota_k$ and $\delta_k$ denote the increasing and decreasing diagonal patterns of size $k$, respectively. Skew shape conjecture. For any skew shape $S$ and any $k\geq 1$,

$$\operatorname{Tr}(S,\iota_k)\geq\operatorname{Tr}(S,\delta_k).$$

This extends the equality known for Ferrers shapes and for D-free skew shapes, while the inequality is motivated by computational evidence and remains open for arbitrary skew shapes.