Equivariant rigidity conjecture for convex co-compact diagonal actions

Let $(X_1,d_1)$ and $(X_2,d_2)$ be negatively curved Hadamard manifolds, and let $\Gamma$ be a torsion-free group acting properly discontinuously and co-compactly on $X_1$ and $X_2$. Suppose the diagonal action of $\Gamma$ on $X_1\times X_2$ is convex co-compact. Equivariant rigidity conjecture. There exists a $\Gamma$-equivariant isometry

$$\phi:(X_1,d_1)\to (X_2,\lambda d_2)$$

for some $\lambda>0$. This statement is presented as a consequence that would follow from a positive answer to the Burns–Katok marked length spectrum conjecture, and its resolution is not established in the supplied source.