On Auroux's Luttinger-surgery uniqueness question

We construct simply connected integral symplectic four-manifolds forming an exotic pair, with equal values of $c_1^2$, $c_2$, $c_1\mathbin{\cdot}[ω]$, and $[ω]^2$, but with symplectic Kodaira dimensions $-\infty$ and $2$. Since symplectic Kodaira dimension is invariant under Luttinger surgery, the t