The finite-index Anosov representation conjecture for cyclic doubles

Let $\Gamma$ be an Anosov group and let $\langle w\rangle$ be a maximal cyclic subgroup of $\Gamma$. A finite-index subgroup $\Gamma'$ of $\Gamma$ containing $w$ is said to be suitable for the double if the amalgamated product $\Gamma'\ast_{\langle w\rangle}\Gamma'$ admits an Anosov representation.

Finite-index Anosov representation conjecture. There exists a finite-index subgroup $\Gamma'$ of $\Gamma$ containing $w$ such that

$$\Gamma'\ast_{\langle w\rangle}\Gamma'$$

admits an Anosov representation.

This is presented as a weaker statement than the question of whether the double of $\Gamma$ itself is Anosov, possibly in a larger group. The source states that the authors strongly believe this weaker assertion, but provides no resolution.