Rolland Rigidity conjecture for self-diffeomorphisms of locally symmetric spaces

Suppose $M$ is a locally symmetric space with natural Riemannian metric $\rho$ and Levi-Civita connection $abla$. Let $f$ be a self-diffeomorphism of $M$, and let $(g_{1,b})$ and $(g_{2,b})$ be distinct smooth root systems to the identity inducing distinct vector fields $\xi_1$ and $\xi_2$. Rolland Rigidity conjecture. There is a class of such self-diffeomorphisms for which, for every $p\in M$, the points $p$ and $f(p)$ are conjugate with respect to $abla$. For each self-diffeomorphism in this class, there should be a differential equation $\xi_b$ on $M$, positive integers $k_1,k_2$, and inversion symmetries $P_1,P_2$ of $(M,\rho)$ satisfying

$$k_iDP_i\circ\xi_c=\xi_i\circ P_i.$$

Here $DP_i:TM\to TM$ is the induced map on the tangent bundle. If $abla$ is nonpositively curved, then $(M,\nabla)$ has no conjugate points, and every self-diffeomorphism in this class should induce a unique differential equation $\xi$ and hence a necessarily unique flow $\Phi_t$ with

$$f=\Phi_{t=1}.$$

The claim proposes a rigidity phenomenon determined by $f$, with the base differential equation $\xi_b$ serving as part of that structure. Its status is unclear from the supplied text; the source presents it as a suspicion rather than establishing it as a theorem.