Equivalence of quantizations of the dispersionless KdV hierarchy

Wang recently constructed a quantization of the dispersionless KdV hierarchy using the Heisenberg vertex algebra. Independently, in joint work with Rossi, we obtained a quantization of the dispersionless KdV hierarchy as the trivial Cohomological Field Theory case of the meromorphic differential hierarchies. In this note, we prove that the two constructions coincide.