  In [41] Okuyama and Sakai gave a conjectural equality for the higher genus
generalized Br\'ezin--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 ${\it ibid}$. 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.
