Residue-class unimodality conjecture for rational Catalan polynomials

Let $a<b$ be relatively prime, and write

$$\mathbf{Cat}_{a,b}(q)=\sum_j c_jq^j.$$

For each residue class $r$ with $0\leq r<a$, consider the coefficient sequence $(c_{ak+r})_k$ indexed by the integers $k$ for which these coefficients occur.

Residue-class unimodality conjecture. For every residue class $r$, $0\leq r<a$, the coefficients of $q^{ak+r}$ in $\mathbf{Cat}_{a,b}(q)$ are unimodal. The conjecture refines ordinary unimodality, which the source notes does not hold for the coefficients of $\mathbf{Cat}_{a,b}(q)$ in general. It is motivated by the shifted-coset formula above, but no resolution is supplied.