The automorphism-extension conjecture for models of normalizers in fusion systems

Let $\calf$ be a saturated fusion system over a $p$-group $S$ such that $O_p(\calf)=1$. Set $\cale=F^*(\calf)$, a fusion subsystem over $T\nsg S$, and let $\call$ be a centric linking system associated to $\cale$. Set $\Gamma_0=\operatorname{Aut}_{\call}(T)$, a model for $N_{\cale}(T)$. Let $\Gamma$ be a model for $N_{\calf}(T)$, and identify $\Gamma_0$ with a normal subgroup of $\Gamma$. Write $\operatorname{Aut}^{I}(\call)$ for the group of isotypical automorphisms of $\call$. Automorphism-extension conjecture. There is a homomorphism $\tau:\Gamma\to\operatorname{Aut}^{I}(\call)$ making both triangles in the diagram commute, namely its restriction to $\Gamma_0$ is conjugation and the induced automorphism of $\Gamma_0$ agrees with conjugation by the corresponding element of $\Gamma$:

$$\begin{CD} \Gamma_0 @>{\operatorname{conj}}>> \operatorname{Aut}^{I}(\call)\\ @V{\operatorname{incl}}VV @VV{\alpha\mapsto\alpha_T}V\\ \Gamma @>{\operatorname{conj}}>> \operatorname{Aut}(\Gamma_0) \end{CD}$$

This is intended to provide the compatibility needed to construct extensions of fusion systems from normalizer models; the source gives no resolution.