Polynomiality conjecture for faithful dimensions over finite truncated valuation rings

Let $\mathfrak{g}$ be a nilpotent $\mathbb{Z}$-Lie algebra finitely generated as an abelian group. For a finite truncated valuation ring $R$ with maximal ideal $\mathfrak{m}$, write $d$ for the least positive integer with $\mathfrak{m}^d=0$, let $p=\operatorname{char}(R/\mathfrak{m})$, let $pR=\mathfrak{m}^e$, and write $R/\mathfrak{m}\cong\mathbb{F}_{p^f}$; call $(d,p,e,f)$ its associated parameters. Let $M(\mathfrak{g})$, partitions $\{\mathscr{P}_1,\ldots,\mathscr{P}_r\}$ and $\{\mathscr{F}_1,\ldots,\mathscr{F}_s\}$, and polynomials $g_{ij}(T)$ be as in the polynomiality theorem for $m_{\mathrm{faithful}}(\mathscr{G}_{\mathbb{F}_q})$. Polynomiality conjecture. These data can be chosen such that, for every finite truncated valuation ring $R$ with associated parameters $(d,p,e,f)$,

$$m_{\mathrm{faithful}}(\mathscr{G}_R)=f\sum_{\ell=0}^{e-1}g_{ij}\bigl(p^{f(d-\ell)}\bigr)$$

whenever $(p,f)\in\mathscr{P}_i\times\mathscr{F}_j$. This conjecture extends the polynomiality result over finite fields to finite truncated valuation rings, asserting that the faithful dimension depends on the ring through the associated parameters in the specified polynomial form.