Non-geometric 3-manifold virtual embedding conjecture

Let $M$ be a non-geometric closed aspherical $3$-manifold. A closed essentially immersed subsurface of $M$ is virtually embedded if it lifts to an embedded subsurface in some finite cover of $M$.

Virtual embedding conjecture. Every non-geometric closed aspherical $3$-manifold contains a closed essentially immersed subsurface which is not virtually embedded.

This conjecture proposes a general obstruction to virtual embedding for immersed subsurfaces in non-geometric $3$-manifolds. The examples preceding it establish such obstructions for mapping tori with nontrivial fractional Dehn twist coefficients, but the statement for all non-geometric closed aspherical $3$-manifolds remains open.