Fixed-point conjecture for Grassmannian braid-group actions

Let $d=\gcd(k,n)>1$, assume $n=k+\ell+1$, and assume $\mathbb{C}[\operatorname{Gr}(k,n)]$ is of infinite type. Let $\operatorname{Br}_d$ be the braid group with generators $\sigma_i$ for $i\in[d-1]$. A fixed point, stable fixed point, unstable fixed point, cluster monomial, and prime non-real element are understood as defined in the source.

Braid-group fixed-point conjecture. For every $i\in[d-1]$, if $(k,n)$ is $(3,9)$ or $(4,8)$, every unstable fixed point of $\sigma_i$ in $\mathbb{C}[\operatorname{Gr}(k,n)]$ is a cluster monomial; and for every $(k,n)$ under the stated hypotheses, every stable fixed point of $\sigma_i$ is prime non-real.

This combines two claims about the stable and unstable fixed points of braid-group generators. The supplied text gives no resolution status.