Amdeberhan's generating-function conjecture for the maximum number of 1-hooks

Let $P(n)$ denote the set of partitions of $n$, and let $\alpha_1(\lambda)$ be the number of $1$-hooks in a partition $\lambda$, equivalently the number of distinct parts. Define

$$b_n=\max\{\alpha_1(\lambda)\colon \lambda\in P(n)\}.$$

Write

$$(q;q)_\infty=(1-q)(1-q^2)(1-q^3)\cdots.$$

Amdeberhan's conjecture. The generating function for the sequence $b_n$ is

$$\sum_{n\geq 0}b_nq^n=\frac{1}{1-q}\left(\frac{(q^2;q^2)_\infty^2}{(q;q)_\infty}-1\right).$$

The paper states that this conjecture is proved, so the claim is recorded as solved rather than open.