Refined floor diagrams relative to a conic and Caporaso-Harris type formula

We prove a $q$-refined correspondence theorem between higher genus relative Gromov-Witten invariants with a Lambda class $\lambda_{g-g'}$ insertion in the blow-up of $\mathbb{P}^2$ at $k$ points on a conic and the refined counts of genus $g'$ floor diagrams relative to a conic, after the change of variables $q=e^{iu}$. We provide a Caporaso-Harris type recursive formula for the refined counts of higher genus floor diagrams. As an application of the correspondence theorem, we propose a higher genus version of the BPS polynomials of del Pezzo surfaces of degree $\geq3$ and Hirzebruch surfaces, which generalize the higher genus Block-G\"ottsche polynomials.