Solenoidal injectivity conjecture for locally symmetric metrics

Let $g_0$ be a locally symmetric metric, and let $D_{g_0}^*$ be the operator appearing in the paper. Denote by $\Pi_{\operatorname{Ker}(D_{g_0}^*)}$ the projection onto $\operatorname{Ker}(D_{g_0}^*)$, and let $Q$ be the operator defined in the paper. Solenoidal injectivity conjecture. The operator

$$\Pi_{\operatorname{Ker}(D_{g_0}^*)}Q$$

is solenoidal injective for any locally symmetric metric.

The source proves solenoidal injectivity for real and complex hyperbolic metrics and identifies the remaining locally symmetric cases as a geometric and dynamical problem; the general assertion remains open.