Lyapunov-spectrum characterization of negatively curved locally symmetric spaces

Let $\mathbf{H}$ be a negatively curved symmetric space with $\dim \mathbf{H} \geq 3$. For a closed negatively curved Riemannian manifold $Y$, let $g^t_Y$ be its geodesic flow, let $p$ be a periodic point of $g^t_Y$, and let $\nu^{(p)}$ denote the invariant measure associated with the periodic orbit of $p$. Write $\vec{\lambda}(\mathbf{H})$ for the common Lyapunov spectrum of the geodesic flows of all negatively curved locally symmetric spaces with universal cover $\mathbf{H}$.

Lyapunov-spectrum characterization conjecture. Let $Y$ be a closed negatively curved Riemannian manifold, $\dim Y \geq 3$. Suppose that for each periodic point $p$ of $g^{t}_{Y}$ there exists a constant $\omega(p) > 0$ such that

$$\vec{\lambda}(g_{Y},\nu^{(p)}) = \omega(p)\vec{\lambda}(\mathbf{H}).$$

Then $Y$ is locally symmetric with universal cover homothetic to $\mathbf{H}$.

The conjecture is motivated by the paper's periodic-orbit rigidity theorem, which proves a corresponding characterization under the stated spectral hypotheses in the locally symmetric setting; the general assertion for negatively curved manifolds remains open.