The bivariate symmetry 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. Let $s\ell(\lambda)$ be its skew length, and define its co-skew length by

$$s\ell'(\lambda)=\frac{(a-1)(b-1)}{2}-s\ell(\lambda).$$

Bivariate symmetry conjecture.

$$\sum_{\lambda}q^{\ell(\lambda)}t^{s\ell'(\lambda)}=\sum_{\lambda}t^{\ell(\lambda)}q^{s\ell'(\lambda)},$$

where both sums are over $(a,b)$-cores $\lambda$. This is presented as a shadow of the broader $q,t$-Catalan theory and is expected to be difficult; no resolution is supplied in the source.