On a new proof of the Okuyama--Sakai conjecture

In [41] Okuyama and Sakai gave a conjectural equality for the higher genus generalized Brézin–Gross–Witten (BGW) free energies. In a recent work [46] we established the Hodge-BGW correspondence on the relationship between certain special cubic Hodge integrals and the generalized BGW correlators, and a proof of the Okuyama–Sakai conjecture was also given $ibid$. In this paper, we give a new proof of the Okuyama–Sakai conjecture by a further application of the Dubrovin–Zhang theory for the KdV hierarchy.