Ghiggini's conjecture on separating half Giroux torsion

Let $(Y,\xi)$ be a closed contact 3-manifold with half Giroux torsion along a separating torus. Its contact invariant is an element

$$\widehat{c}(\xi)\in\widehat{\mathit{HF}}(-Y).$$

Ghiggini's conjecture. The contact invariant vanishes:

$$\widehat{c}(\xi)=0.$$

The conjecture asserts that separating half Giroux torsion obstructs symplectic fillability. The paper's abstract states that it provides counterexamples, so the conjecture is disproved.