Invariance of involutive bordered Floer modules

Let $Y$ be a bordered $3$-manifold with bordered Heegaard diagram $\mathcal{H}$. The involutive type $D$ structure $\widehat{\mathit{CFDI}}(\mathcal{H})$ and involutive $A_\infty$-module $\widehat{\mathit{CFAI}}(\mathcal{H})$ are defined using the involutive structures and the associated Heegaard-move maps described above. Involutive bordered Floer invariance conjecture. The involutive type $D$ structure $\widehat{\mathit{CFDI}}(\mathcal{H})$ and involutive $A_\infty$-module $\widehat{\mathit{CFAI}}(\mathcal{H})$ are invariants of the bordered $3$-manifold $Y$. This asserts independence of the bordered Heegaard diagram and of the choices of Heegaard moves used to define the involutive structures.