Let $(H_\mathbb{Z}, \{h^{p,q}\}_{p+q=n}, \phi)$ be the datum to define weight $n$ polarized Hodge structures on $H_\mathbb{Z}$, $h^{p,q}$ is the Hodge numbers, $\phi$ is a polarization. Let $D$ be the classifying space of polarized Hodge structures of the above type, and $\check{D}$ the compact dual of $D$. Let $G_R=Aut(H_R, \phi)$ with $R=\mathbb{Q}, \mathbb{R}, \mathbb{C}$, and $\mathfrak{g}_R:=Lie(G_R)$ the associated Lie algebra. Let $N\in \mathfrak{g}_\mathbb{Q}$ be a nilpotent element whose index at most equals to rank$H_\mathbb{Z}$. Due to the fundamental works of W. Schmid et al., we may say a nilpotent orbit w.r.t. $N$ is a map on the upper-half plane $\mathfrak{h}$, together with a fixed $F\in \check{D}$ \begin{equation} \theta: \mathfrak{h}\rightarrow \check{D}, ~\theta(z)=\operatorname{exp}(zN)F \end{equation} satisfying

1. $NF^p\subset F^{p-1}$;
2. $\operatorname{exp}(zN)F\in D, \operatorname{Im}z\to \infty$.

For a nilpotent orbit as above, the monodromy weight filtration $W(N)$ defined by $N$ and the decreasing filtration $F$ determines a polarized mixed Hodge structure on $H_\mathbb{Z}$.

My question: for a fixed nilpotent element $N$, the associated polarized MHS's to all $N$-nilpotent orbits are of the same type? More precisely, if $F\in \check{D}$ satisfies the nilpotent orbit condition as above, are the virtual Hodge numbers \begin{equation} h^{p,q}_N:=F^pGr^W_k\cap \overline{F^p}Gr^W_k \end{equation} of the mixed Hodge structure $(W(N), F)$ invariant?

Let $(H_\mathbb{Z}, \{h^{p,q}\}_{p+q=n}, \phi)$ be the datum to define weight $n$ polarized Hodge structures on $H_\mathbb{Z}$, $h^{p,q}$ is the Hodge numbers, $\phi$ is a polarization. Let $D$ be the classifying space of polarized Hodge structures of the above type, and $\check{D}$ the compact dual of $D$. Let $G_R=Aut(H_R, \phi)$ with $R=\mathbb{Q}, \mathbb{R}, \mathbb{C}$, and $\mathfrak{g}_R:=Lie(G_R)$ the associated Lie algebra. Let $N\in \mathfrak{g}_\mathbb{Q}$ be a nilpotent element whose index at most equals to rank$H_\mathbb{Z}$. Due to the fundamental works of W. Schmid et al., we may say a nilpotent orbit w.r.t. $N$ is a map on the upper-half plane $\mathfrak{h}$, together with a fixed $F\in \check{D}$ \begin{equation} \theta: \mathfrak{h}\rightarrow \check{D}, ~\theta(z)=\operatorname{exp}(zN)F \end{equation} satisfying

1. $NF^p\subset F^{p-1}$;
2. $\operatorname{exp}(zN)F\in D, \operatorname{Im}z\to \infty$.

For a nilpotent orbit as above, the monodromy weight filtration $W(N)$ defined by $N$ and the decreasing filtration $F$ determines a polarized mixed Hodge structure on $H_\mathbb{Z}$.

My question: for a fixed nilpotent element $N$, the associated polarized MHS's to all $N$-nilpotent orbits are of the same type? More precisely, if $F\in \check{D}$ satisfies the nilpotent orbit condition as above, are the virtual Hodge numbers \begin{equation} h^{p,q}_N:=F^pGr^W_k\cap \overline{F^{k-p}}Gr^W_k \end{equation} of the mixed Hodge structure $(W(N), F)$ invariant?