The classification conjecture for square-zero sphere classes

Let $X$ be the smooth, closed, oriented $4$-manifold under consideration, with signature $\sigma$ and modified Euler characteristic $\tilde \chi$, and let $\Gamma$ denote its intersection lattice. Let $A$ be a homology class with $A\cdot A=0$ represented by an embedded sphere. Write $\operatorname{Aut}(\Gamma)^T$ for the subgroup of lattice automorphisms preserving the relevant $T$-structure. Square-zero sphere-class conjecture. If

$$2\tilde \chi+3\sigma<0,$$

then $A$ is equivalent under $\operatorname{Aut}(\Gamma)^T$ to one of the classes in the last part of the sphere-class proposition. The claim extends the preceding classification to the case $2\tilde \chi+3\sigma<0$; the paper presents this as a hoped-for classification, and no resolution is supplied in the given text.