Expected action of the type D descent algebra on D_n-Lie monomials

Let $D_n$ be the Weyl group of type $D$ with descent algebra $\Sigma_{D_n}$, let $\mathcal{C}(n)$ denote the relevant class of compositions, and let $P^{(\kappa)}$ and $B_\nu$ be the corresponding type $D$ Lie-monomial and descent-algebra elements. For filled templates $\boldsymbol{z}\in Z(\kappa,\nu)$, write $r(\boldsymbol{z})$ for the associated composition and use tildes for the corresponding type $D$ sums. Expected type D Lie-monomial multiplication property. The action should satisfy

$$P^{(\kappa)}B_\nu=\sum _{\boldsymbol{z}} \tilde{P}^{(r(\boldsymbol{z}))},$$

where the sum is over filled templates with exactly one non-zero entry in each column; moreover, $P^{(\eta)}$ should be a summand of $\tilde{P}^{(r(\boldsymbol{z}))}$ if and only if $B_{(\eta)}$ is a summand of $\tilde{B}_{(r(\boldsymbol{z}))}$. Also, $P^{(\kappa)}B_\nu$ should vanish unless adjacent components of $\kappa$ sum to give the components of $\nu$, excluding the two cases in which $\kappa$ and $\nu$ belong respectively to $\mathcal{C}_n$ and $\mathcal{C}'_n$ (or conversely) and all components are even. This is an expected type $D$ analogue of the established descriptions for the symmetric and hyperoctahedral groups; the conjecture concerns the still-to-be-defined $D_n$-Lie monomials and their compatibility with multiplication in the type $D$ descent algebra.