The polynomiality conjecture for local invariants of p-compact groups

Let $\mathbf{X}$ be a (simply) connected $p$-compact group, let $q$ be a prime power prime to $p$, and let $\mathcal{F}(q)$ be the saturated $p$-fusion system associated to the homotopy fixed-point space $\mathbf{X}(q)$ of the unstable Adams operation $\psi^q$. Let $\alpha$ be a compatible Külshammer–Puig family associated to $\mathcal{F}(q)$. Define

$$k:=\begin{cases} \operatorname{ord}_p(q) & \text{if }p>2,\\ 2 & \text{if }p=2, \end{cases}$$

where $\operatorname{ord}_p(q)$ is the order of $q$ modulo $p$, and set $l=v_p(1-q^k)$.

Polynomiality conjecture. For each $d\geqslant 0$, the invariant $\mathbf{m}(\mathcal{F}(q),\alpha,d)$ is a rational polynomial in $p^l$.

This is proposed as a generalization of the corresponding result for the Benson–Solomon fusion systems. The candidate is not accompanied by a supplied resolution status, so it remains open here.