The sign-character formula for the $(1,3)$-bosonic-fermionic coinvariant ring

Let $R_n^{(1,3)}$ be the $(1,3)$-bosonic-fermionic coinvariant ring, let $\operatorname{Frob}(R_n^{(1,3)};q;u,v,w)$ denote its Frobenius series, and let $s_{(1^n)}$ denote the Schur function of the sign representation. The $(1,3)$ sign-character conjecture. Based on data for $n \leq 5$, the source proposes

$$\left\langle \operatorname{Frob}( R^{(1,3)};q;u,v,w), s_{(1^n)}\right\rangle = \sum_{k,\ell,d \geq 0} u^k v^\ell w^d q^{\binom{n-d-k-\ell}{2}}\qbinom{n-1-d}{\ell}_q\qbinom{n-1-k}{d}_q\qbinom{n-1-\ell}{k}_q.$$

Here $\qbinom{a}{b}_q$ denotes the $q$-binomial coefficient. The formula conjecturally gives the multigraded sign-character multiplicity; the supplied text reports computational evidence for $n \leq 5$ but no resolution.