Positive surface curvature implies no nonpositive-Euler-characteristic surface subgroups

Let $X$ be a finite, standard, irreducible 2-complex, and let $\sigma_-(X)$ denote its negative surface curvature invariant. Let $S$ be a closed surface with $\chi(S)\leq 0$.

Positive surface curvature implies no nonpositive-Euler-characteristic surface subgroups. If $\sigma_-(X)>0$ then no subgroup of $\pi_1(X)$ is isomorphic to $\pi_1(S)$.

This is a conjectural lower-curvature constraint on surface subgroups; the source provides no known cases or resolution.