Invariant stability conditions on local $\mathbb{P}^1\times \mathbb{P}^1$ (after Del Monte-Longhi)

Let $X$ be the total space of canonical bundle of $\pp$, we study an invariant subspace of stability conditions on $X$ under an autoequivalence of $D^b(X)$. We describe the complete set of stable objects with respect to the invariant stability conditions and characterize the space of invariant stability conditions.