Total geodesy conjecture for compact splitting submanifolds of rank-2 type-I quotients

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

Rank-2 type-I total geodesy conjecture. If

$$\dim(S)\geq p,$$

then $(S,g|_S)\hookrightarrow (X,g)$ is totally geodesic.

The preceding theorem establishes total geodesy under the stronger condition $\dim(S)>p$. The conjecture lowers this threshold by one; examples from graphs of non-covering maps show why dimensions below $p$ require care.