The universal orientable 3-manifold embedding conjecture for stabilized $S^2$-bundles

Let $M$ be an orientable $3$-manifold. Universal embedding conjecture. There is a positive integer $n$ such that $M$ embeds almost in the spine of a minimal-genus trisection of

$$\#^n S^2 \times S^2.$$

The paper presents this as a general expectation extending the lens-space constructions; no proof or counterexample is given.