The genus-two classification conjecture for 3-manifolds almost in a trisection spine

Let $M$ be a closed orientable $3$-manifold that is not of the form $\#^k S^1 \times S^2$, and let $M$ embed so as to almost lie in the spine of a genus-$2$ trisection of a $4$-manifold $X$. Genus-two classification conjecture. Then $M$ is a lens space $L(n,1)$, and $X$ is either $S^2 \times S^2$ or $S^2 \tilde{\times} S^2$; moreover, $X$ is $S^2 \times S^2$ when $n$ is even and $S^2 \tilde{\times} S^2$ when $n$ is odd. The preceding construction suggests that this is the only interesting possibility in genus $2$; no resolution is supplied.