The no essential sphere or torus conjecture for the four-manifold $AP$

Let $AP$ be the four-manifold constructed in the paper, and let $H_2(AP;\mathbb Z)$ denote its second homology group. An embedded surface in $AP$ is homologically essential if its homology class is nonzero. The no essential sphere or torus conjecture. The four-manifold $AP$ does not contain homologically essential embedded spheres or tori. This is motivated by an extensive search that found no such representatives; Seiberg–Witten theory rules out the relevant classes of self-intersection at least $-1$, but the claim for more negative squares remains unresolved.