Anosov combination conjecture

Let $G$ be the ambient group, let $P_{{\tau_{\mathrm{mod}}}}$ be the parabolic subgroup associated with the face type $\tau_{\mathrm{mod}}$, and let $A_1,\ldots,A_n\subset \mathrm{Flag}\left(\tau_{\mathrm{mod}}\right)$ be nonempty compact subsets. Assume that any two distinct elements of

$$A:=\bigcup_{i=1}^n A_i$$

are antipodal. For each $i$, let $\Gamma_i$ be a subgroup of $G$.

Anosov combination conjecture. If each $\Gamma_i$ is $P_{{\tau_{\mathrm{mod}}}}$-Anosov and, for every $i=1,\ldots,n$ and every $\gamma\in\Gamma_i-\{1\}$,

$$\gamma\left(A-A_i\right)\subset \operatorname{int}\left(A_i\right),$$

then the subgroup $\Gamma$ of $G$ generated by $\Gamma_1,\ldots,\Gamma_n$ is $P_{{\tau_{\mathrm{mod}}}}$-Anosov.

This is a proposed combination theorem for Anosov subgroups in the spirit of classical Klein--Maskit combination theorems and quantitative ping-pong arguments. Unlike the preceding theorem in the paper, it is stated without passing to finite-index subgroups and uses antipodal compact subsets of the flag variety as the dynamical domains. The supplied text gives no resolution, so the conjecture is recorded as open.