The skew-length generating-function conjecture for core partitions

Let $a<b$ be coprime. For an $(a,b)$-core partition $\lambda$, let $\ell(\lambda)$ be its number of nonzero rows. Define its skew length $s\ell(\lambda)$ to be the number of boxes simultaneously in its $b$-boundary, consisting of boxes with hook lengths less than $b$, and in its $a$-rows, obtained by selecting the highest row in each residue class modulo $a$ among the hook lengths of boxes in the first column. Skew-length generating-function conjecture.

$$\sum_{\lambda} q^{\ell(\lambda)+s\ell(\lambda)}=\frac{1}{[a+b]_q}{a+b \brack a,b}_q,$$

where the sum is over $(a,b)$-cores $\lambda$. This conjecturally supplies a statistic solving the preceding $q$-Catalan problem; its relation to known major-index statistics in the classical Catalan case is unclear, and the source expects the conjecture to be difficult.