Multigraded sign-character formula for R_n^{(1,3)}

Prove that, for every integer n≥1, the q,u,v,w-graded multiplicity of the sign character in the one-boson, three-fermion coinvariant ring R_n^{(1,3)} is \[\left\langle\Frob(R^{(1,3)};q;u,v,w),s_{(1^n)}\right\rangle=\sum_{k,\ell,d\geq0}u^kv^\ell w^d q^{\binom{n-d-k-\ell}{2}}\qbinom{n-1-d}{\ell}_q\qbinom{n-1-k}{d}_q\qbinom{n-1-\ell}{k}_q.\]

Background

The paper derives an upper bound for the sign-character multiplicity in R_n^{(1,3)}, conditional on the t=0 specialization of the Theta conjecture for R_n^{(1,2)}. Computational data for n≤5 suggests that this upper bound is attained.

The displayed identity is proposed as a q,u,v,w-refinement of a numerical conjecture of Bergeron. The paper does not prove the equality in general; it explains that proving the relevant Bergeron conjecture together with the conditional Theta specialization would force equality in the upper bound.