Total geodesy conjecture for compact splitting submanifolds of type-IV quotients

Let $n\geq 3$ be an integer, let $\Gamma\subset \operatorname{Aut}(D^{IV}_n)$ be a torsion-free discrete subgroup, and write $X=D^{IV}_n/\Gamma$. Let $g$ be a canonical Kähler–Einstein metric on $X$, and let $S\subset X$ be a compact splitting complex submanifold with $\dim(S)\geq 2$.

Type-IV total geodesy conjecture. Then

$$(S,g|_S)\hookrightarrow (X,g)$$

is totally geodesic.

The paper proves total geodesy for $\dim(S)>n/\sqrt{2}$ and explains that the conjecture would extend this to every compact splitting complex submanifold of dimension at least $2$. It is motivated by the more rigid behavior expected for the noncompact dual of the hyperquadric.