Extension of the stretched Kostka quasi-polynomial degree formula to arbitrary types

Let $\mathfrak{g}$ be a complex semisimple Lie algebra, with root system $\Phi$, positive roots $\Phi_+$, simple roots $\alpha_1,\ldots,\alpha_r$, and corresponding fundamental weights $\omega_1,\ldots,\omega_r$. Let $\lambda$ and $\mu$ be dominant weights such that

$$\lambda-\mu=\sum_{i=1}^r c_i\alpha_i,\qquad c_i\in\mathbb{Z}_{\geq0},$$

and write $\lambda=\sum_{i=1}^r d_i\omega_i$ with $d_i\in\mathbb{Z}_{\geq0}$. Define $\Phi^{(1)}$ as the root subsystem spanned by the simple roots $\alpha_i$ with $c_i\neq0$, and define $\Phi^{(2)}$ as the root subsystem of $\Phi^{(1)}$ spanned by the simple roots with $c_i\neq0$ and $d_i=0$. For the stretched Kostka quasi-polynomial $K_{\lambda,\mu}(N):=\dim V_{N\lambda}(N\mu)$, the degree is

$$\deg K_{\lambda,\mu}(N)=|\Phi^{(1)}_+|-\operatorname{rk}(\Phi^{(1)})-|\Phi^{(2)}_+|.$$

The preceding formula is established in the source for complex semisimple Lie algebras of classical type; this conjectural extension concerns all types and remains unresolved in the supplied text.