An exotic $S^2\times S^2$ and an exotic $\mathbb{CP}^2\#\overline{\mathbb{CP}}^2$

We prove that a specified Lidman-Piccirillo piece $V$, a symplectic $4$-manifold with the homology of $S^2\times D^2$ built from a genus-$2$ surface bundle over a once-punctured torus by two Luttinger surgeries, is simply connected, for an explicit permitted choice of the two surgery parametrization