A universal linear lower bound for sphere self-intersections

Let $X$ be a closed, oriented four-manifold with $b_2^+(X)>1$ and nontrivial Seiberg–Witten invariant $SW_X$, and let $S\subset X$ be a smoothly embedded sphere. Write $[S]^2$ for its self-intersection and $b_2(X)$ for the second Betti number.

Universal linear bound conjecture. There is a universal constant $C$ such that

$$[S]^2\geq C\cdot b_2(X).$$

The conjecture asks whether sphere self-intersections in four-manifolds with nontrivial Seiberg–Witten invariants admit a uniform linear lower bound in terms of the second Betti number. The paper's preceding discussion identifies boundedness of the possible self-intersections as an open question, while the stated conjecture proposes a stronger quantitative form.