Bergeron's Fibonacci conjecture for the sign character of the $(1,3)$-bosonic-fermionic coinvariant ring

Let $R_n^{(1,3)}$ be the $(1,3)$-bosonic-fermionic coinvariant ring, let $\operatorname{Frob}(R_n^{(1,3)};1;1,1,1)$ denote its Frobenius series with all grading variables specialized to $1$, and let $s_{(1^n)}$ be the Schur function of the sign representation. Let $F_n$ be the Fibonacci numbers defined by $F_0=0$, $F_1=1$, and $F_n=F_{n-1}+F_{n-2}$ for $n\geq2$. Bergeron's Fibonacci conjecture. For all $n \geq 1$,

$$\left\langle \operatorname{Frob}(R_n^{(1,3)};1;1,1,1), s_{(1^n)}\right\rangle = \frac{1}{2}F_{3n}.$$

This conjecture predicts a Fibonacci-number formula for the multiplicity of the sign character after specialization. The source attributes it to Bergeron and reports computational evidence, but does not specify a resolution.