Generalized M-polynomial product conjecture for graded Lie algebra components

Let $\mathfrak g=\bigoplus_{i\in\mathbb Z}\mathfrak g(i)$ be a $\mathbb Z$-grading of a simple Lie algebra, and let $\Delta(1)$ be the corresponding weight poset with height function ${\mathsf{ht}}$. Let $\mathfrak{An}(\Delta(1))$ denote its antichains and let $\EuScript M_{\Delta(1)}(t)$ be its M-polynomial.

Generalized M-polynomial product conjecture. One has

$$\EuScript M_{\Delta(1)}(t)=\prod_{\gamma\in\Delta(1)}\frac{1-t^{{\mathsf{ht}}(\gamma)+1}}{1-t^{{\mathsf{ht}}(\gamma)}}.$$

In particular,

$$\#\mathfrak{An}(\Delta(1))=\EuScript M_{\Delta(1)}(1)=\prod_{\gamma\in\Delta(1)}\frac{{\mathsf{ht}}(\gamma)+1}{{\mathsf{ht}}(\gamma)}.$$

The formula is proved in the abelian and extra-special cases and in some additional small product cases, but remains open for arbitrary gradings.