Characterization of products with compressed fixed subgroups of automorphisms

Let $G=G_1\times\cdots\times G_n$, where $n\geqslant1$, each $G_i$ is a finitely generated free group or a surface group, and a subgroup $H\leqslant G$ is compressed in $G$ if $\mathrm{rk}(H)\leqslant\mathrm{rk}(K)$ for every finitely generated subgroup $K$ satisfying $H\leqslant K\leqslant G$.

Compressed fixed-subgroup characterization. The subgroup $\mathrm{Fix}\,\phi$ is compressed in $G$ for every $\phi\in\operatorname{Aut}(G)$ if and only if $G$ is of one of the forms (euc1), (euc2), (euc3), (euc4), (hyp1), (hyp2), or (hyp3).

The source presents this as a conjecture in the discussion of the Euclidean cases, and notes that proving the related endomorphism inertia conjectures would imply a positive solution to a broader statement; the resolution status is not specified.