Question:Consider the product Riemannian manifold $(M,g)=(S^n\times H^k,g_{round}\oplus g_{hyp})$ of a round sphere $(S^n,g_{round})$ of curvature $1$ and hyperbolic space $(H^k,g_{hyp})$ of curvature $-1$. Is it true that the only conformal transformations of $(M,g)$ are isometries? In other words, is $\mathrm{Conf}(M,g)=\mathrm{Iso}(M,g)$?

Although I am mainly interested in the case $n>k>1$, it would also be nice to know the answer for all $n,k\geq2$. Note that by a result of Ferrand (see Schoen), the action of $\mathrm{Conf}(M,g)$ is proper, hence there exists a metric $g_*$ conformal to $g$, such that $\mathrm{Conf}(M,g)=\mathrm{Iso}(M,g_*)$.