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 λg-g' insertion in the blow-up of P² 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=eⁱᵘ. 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 ≥3 and Hirzebruch surfaces, which generalize the higher genus Block-Göttsche polynomials.