A shifted-coset formula for rational Catalan polynomials

Let $a<b$ be relatively prime. Let $\boldsymbol{\Lambda}$ be the lattice of $a$-cores, let $\boldsymbol{\Lambda}_T=(a\mathbb{Z})^{a-1}$, and let $\mathfrak{c}$ range over the cosets in $\boldsymbol{\Lambda}/\boldsymbol{\Lambda}_T$. For each coset, let $s(\mathfrak{c},b)$ denote the shift determining the intersection of the coset with the simplex of $(a,b)$-cores, and let $\mathcal{SC}_{a-1}(b)$ denote that simplex.

Coset age conjecture. There is an age function $\iota$ on the cosets $\mathfrak{c}\in\boldsymbol{\Lambda}/\boldsymbol{\Lambda}_T$ such that

$$\sum_{\mathfrak{c}\in\boldsymbol{\Lambda}/\boldsymbol{\Lambda}_T}q^{\iota(\mathfrak{c})}=[a]_{q^2}[a]_{q^3}\cdots[a]_{q^{a-1}}$$

and

$$\mathbf{Cat}_{a,b}(q)=\sum_{\mathfrak{c}\in\boldsymbol{\Lambda}/\boldsymbol{\Lambda}_T}q^{\iota(\mathfrak{c})}\binom{b/a-s(\mathfrak{c},b)+a-1}{a-1}_{q^a},$$

where the $q^a$-binomial coefficient $q^a$-counts the points in $\mathfrak{c}\cap\mathcal{SC}_{a-1}(b)$. This would provide a shifted lattice-point expression for the rational Catalan polynomial and explain the positivity of its coefficients. The source presents this as conjectural and gives no resolution.