Flat braid groups, right-angled Artin groups, and commensurability

For every $n\ge 1$, the flat braid group ${\mathrm{F}\mathrm{B}}_n$ is an analogue of the braid group $B_n$ that can be described as the fundamental group of the configuration space $$\{\{x_1,\dots ,x_n\}\in {\mathbb{R}}^n\mathrm{/}\mathrm{S}\mathrm{y}\mathrm{m}(n)\mid \text{there exist at most two indices }i,j\text{ such that }{x}_i=x_j\}\mathrm{.}$$ Alternatively, ${\mathrm{F}\mathrm{B}}_n$ can also be described as the right-angled Coxeter group $C(P_{n-2}^{\mathrm{o}\mathrm{p}\mathrm{p}})$, where $P_{n-2}^{\mathrm{o}\mathrm{p}\mathrm{p}}$ denotes the opposite graph of the path $P_{n-2}$ of length $n-2$. In this article, we prove that, for every $n=7$ or $\ge 11$, ${\mathrm{P}\mathrm{F}\mathrm{B}}_n$ is not virtually a right-angled Artin group, disproving a conjecture of Naik, Nanda, and Singh. In the opposite direction, we observe that ${\mathrm{F}\mathrm{B}}_7$ turns out to be commensurable to the right-angled Artin group $A(P_4)$.