Specialized Theta conjecture at t=0

Establish the specialized Theta-conjecture identity for the (1,2)-bosonic-fermionic coinvariant ring R_n^{(1,2)} given by \(\Frob(R_n^{(1,2)};q;u,v)=\sum_{k+\ell<n}u^kv^\ell\left(\Theta_{e_k}\Theta_{e_\ell}\nabla e_{n-k-\ell}\right)|_{t=0}\) for every n\geq1.

Background

This conjecture is the specialization at t=0 of the recalled Theta conjecture. It concerns the Frobenius series of the coinvariant ring with one bosonic variable set and two fermionic variable sets.

The paper invokes it conditionally to obtain formulas for hook-character multiplicities in R_n^{(1,2)} and an upper bound for the sign-character multiplicity in R_n^{(1,3)}. The paper does not establish the specialized identity.