Explicit formula conjecture for truncated Ehrhart-like polynomials

Let $\Phi$ be a root system with positive roots $\Phi^+$, weight lattice $P$, and root lattice $Q$. Let $L^{\mathrm{tr}}_{\lambda}(\mathbf{k})$ count the weights whose truncated interval-firing stabilization under a good parameter $\mathbf{k}\in\mathbb{N}[\Phi]^W$ is $\lambda$. For a linearly independent subset $X\subseteq\Phi^+$, write $\mathrm{rVol}_Q(X)$ for its relative volume and $\mathbf{k}^{X}$ for the corresponding monomial.

Truncated Ehrhart-like polynomial conjecture. For every $\lambda\in P$ and every good $\mathbf{k}\in\mathbb{N}[\Phi]^W$,

$$L^{\mathrm{tr}}_{\lambda}(\mathbf{k})=\sum_X \mathrm{rVol}_Q(X)\,\mathbf{k}^{X},$$

where the sum runs over all $X\subseteq\Phi^+$ such that $X$ is linearly independent and

$$\langle\lambda,\alpha^\vee\rangle\in\{0,1\}$$

for every $\alpha\in\Phi^+\cap\operatorname{Span}_{\mathbb{R}}(X)$.

The formula is motivated by the established formula for the symmetric Ehrhart-like polynomials and by the decomposition of symmetric fibers into truncated fibers. Its validity for all good parameters and weights remains open.