The non-centric $K$-normalizer conjecture for minimal characteristic bisets

Let $\mathcal F$ be a saturated fusion system on a finite $p$-group $S$, let $\Omega=\Lambda_{\mathcal F}$ be its minimal characteristic biset, and let $P\leq S$ and $K\leq\operatorname{Aut}(P)$ such that $K$ contains or is contained in $\operatorname{Inn}(P)$. Let $\mathcal N$ denote the corresponding $K$-normalizer fusion system. The non-centric $K$-normalizer conjecture. If $P$ is fully $K$-normalized in $\mathcal F$, then

$$N_{\Omega}^{K}(P)=\Lambda_{\mathcal N},$$

without requiring $P$ to be $\mathcal F$-centric. The preceding theorem proves the analogous conclusion with an additional centricity hypothesis, while this conjecture asserts that minimal characteristic bisets still model the normalizer fusion system in the non-centric case. The supplied text gives no resolution status.