Hivert–Mallet–Mallet vectorial conjecture for irreducible k-shapes and surjective pistols

For $k\geq 3$, let $IS_k$ be the set of irreducible $k$-shapes. For an irreducible $k$-shape $\lambda$, let $\overrightarrow{fr}(\lambda)=(t_1,\ldots,t_{k-2})\in\{0,1\}^{k-2}$ record its free $k$-sites. Let $SP_{k-1}$ be the set of surjective pistols, and for $f\in SP_{k-1}$ let $\overrightarrow{fix}(f)=(t_1,\ldots,t_{k-2})\in\{0,1\}^{k-2}$, where $t_i=1$ exactly when $f(2i)=2i$. Mallet's vectorial conjecture. For every $k\geq 3$ and every $\overrightarrow{v}=(v_1,\ldots,v_{k-2})\in\{0,1\}^{k-2}$, the number of $\lambda\in IS_k$ with $\overrightarrow{fr}(\lambda)=\overrightarrow{v}$ equals the number of $f\in SP_{k-1}$ with $\overrightarrow{fix}(f)=\overrightarrow{v}$. This refines the corresponding equality of generating functions by preserving the full statistic vector.