Topological index three conjecture for common Heegaard stabilizations

Let $M$ be a 3-manifold containing unstabilized Heegaard surfaces $F$ and $G$, neither of which has topological index $1$. Suppose that the minimal-genus common stabilization of $F$ and $G$ does not have topological index $2$. Topological index three conjecture. Then $M$ contains a surface of topological index $3$.

The source says that such a manifold would be Haken, so this conjecture complements the preceding question about non-Haken 3-manifolds containing surfaces of topological index at least $3$.