Large-area rigidity conjecture for minimal surfaces in hyperbolic 3-manifolds

Let $M$ be a closed hyperbolic $3$-manifold, and let $\Sigma$ be a minimal surface in $M$. Call $\Sigma$ rigid if there is no metric $g$ with $K_g\leq -1$ in which $\Sigma$ has constant curvature $-1$.

Large-area rigidity conjecture. There exists $A=A(M)$ such that every minimal surface $\Sigma$ with area larger than $A$ is rigid.

This question concerns whether sufficiently large minimal surfaces in a fixed closed hyperbolic $3$-manifold must resist realization as constant-curvature totally geodesic surfaces under metrics with sectional curvature at most $-1$. The source lists it among questions and future directions, and gives no resolution.