Structure of higher-genus open-closed Gromov--Witten theory of $\mathcal{O}_{\mathbb{P}^{1}}(p-1)\oplus\mathcal{O}_{\mathbb{P}^{1}}(-p-1)$

We study the closed and open Gromov--Witten potentials of the toric Calabi--Yau threefold $$ X_p=\operatorname{Tot}\bigl(\mathcal{O}_{\mathbb{P}^{1}}(p-1)\oplus\mathcal{O}_{\mathbb{P}^{1}}(-p-1)\bigr),\qquad p\geq 2. $$ We prove closed and open mirror symmetry under a nonvanishing condition on the t