Bergeron’s Fibonacci conjecture for the sign character

Prove that, for every integer n≥1, the ungraded sign-character multiplicity of the one-boson, three-fermion coinvariant ring R_n^{(1,3)} equals one half of the 3n-th Fibonacci number: \[\left\langle\Frob(R_n^{(1,3)};1;1,1,1),s_{(1^n)}\right\rangle=\frac12F_{3n}.\]

Background

Bergeron proposed a numerical formula for the dimension of the sign-isotypic component of R_n^{(1,3)}, where F_n denotes the Fibonacci sequence with F_0=0, F_1=1, and F_n=F_{n-1}+F_{n-2}.

The paper proves a binomial-sum identity showing that the specialization at q=u=v=w=1 of the paper’s proposed multigraded conjecture equals \frac12F_{3n}. However, that identity does not prove that the actual sign-character multiplicity equals the proposed sum, so Bergeron’s conjecture remains unresolved in the paper.