A conjecture of Han on 3-cores and modular forms

In his study of Nekrasov-Okounkov type formulas on "partition theoretic" expressions for families of infinite products, Han discovered seemingly unrelated $q$-series that are supported on precisely the same terms as these infinite products. In earlier work with Ono, Han proved one instance of this occurrence that exhibited a relation between numbers $a(n)$ that are given in terms of hook lengths of partitions, with numbers $b(n)$ that equal the number of 3-core partitions of $n$. Recently Han revisited the $q$-series with coefficients $a(n)$ and $b(n)$, and numerically found a third $q$-series whose coefficients appear to be supported on the same terms. Here we prove Han's Conjecture about this third series by proving a general theorem about this phenomenon.