Rigidity of convex co-compact diagonal actions

Kleiner-Leeb and Quint showed that convex subsets in higher-rank symmetric spaces are very rigid compared to rank 1 symmetric spaces. Motivated by this, we consider convex subsets in products of proper CAT(0) spaces $X_1\times X_2$ and show that for any two convex co-compact actions ${\rho }_i(\mathrm{\Gamma })$ on $X_i$, where $i=1,2$, if the diagonal action of $\mathrm{\Gamma }$ on $X_1\times X_2$ via $\rho =({\rho }_1,{\rho }_2)$ is also convex co-compact, then under a suitable condition, ${\rho }_1(\mathrm{\Gamma })$ and ${\rho }_2(\mathrm{\Gamma })$ have the same marked length spectrum.