The freeness conjecture for GPTW-generated Lie subalgebras

Let $\mathcal{K}$ be a simplicial complex on $[m]$, let $\mathbf{k}$ be a commutative ring with unit, and define

$$L_\mathcal{K}:=\operatorname{FL}_\mathbf{k}(\mu_1,\dots,\mu_m)/([\mu_i,\mu_j]=0,\ \forall\{i,j\}\in\mathcal{K}).$$

Let $N_\mathcal{K}:=\langle\text{\it GPTW\/}\rangle_{\operatorname{Lie}}\subset L_\mathcal{K}$ be the Lie subalgebra generated by the Grbić–Panov–Theriault–Wu generators, and let $(N_\mathcal{K})_\alpha$ denote its multidegree-$\alpha$ component. The freeness conjecture. The module $N_\mathcal{K}$ is free over $\mathbf{k}$, with

$$(N_\mathcal{K})_\alpha\simeq\mathbf{k}^{\oplus n_\alpha(\mathcal{K})}$$

for every $\alpha\in\mathbb{Z}_{\geq 0}^m$; moreover, an analogue of the stated freeness and chordality theorem holds. The conjecture extends the proved $\mathbb{Z}_2$ flag-complex result and predicts a uniform module-theoretic description over arbitrary commutative coefficient rings; the source does not establish the claimed analogue in this generality.