Bound on the negative self-intersection of divisible embedded surfaces

Let $X$ be a simply-connected $4$-manifold, and let $B$ be an embedded surface in $X$ of genus $g_B$ with self-intersection

$$B^2=-M<0.$$

Assume that the class $[B]$ is divisible by $2$. Embedded-surface self-intersection conjecture. There is a number $c(X)$ depending only on $X$ and a number $\kappa<4$ such that

$$M\leq c(X)+\kappa g_B.$$

This conjecture predicts a linear upper bound on the magnitude of the negative self-intersection in terms of the genus for embedded surfaces whose homology class is divisible by $2$. The supplied text gives the conjecture as motivated by preceding results but provides no evidence that it has been proved or disproved.