Murai–Nevo's multidegree algebra conjecture for the cd-index

Let $\mathbf{k}$ be a field, let $P$ be a Gorenstein$^*$ poset of rank $n+1$, and let $M$ be a degree-$n$ $cd$-monomial. Define the multidegree of $M$ by replacing each $c$ with $0$ and each $d$ with $10$. Murai–Nevo multidegree conjecture. There exists a standard $\mathbb{Z}^n$-graded $\mathbf{k}$-algebra

$$A=\bigoplus_v A_v$$

such that, for every $v\in\mathbb{Z}^n$, the coefficient of the $cd$-monomial $M$ in $\phi_P(c,d)$ with multidegree $v$ is $\dim A_v$. The source states that this is known for posets of Gorenstein-star simplicial complexes but open in general.