Amdeberhan's refined Nekrasov–Okounkov hook length conjecture

Let $\lambda$ be a partition of $n$, and let $\mathcal{H}(\lambda)$ be its multiset of hook lengths. Let $\mathcal{H}(\lambda)^{\diamond}$ be the multiset of hook lengths with trivial legs. Amdeberhan's refinement. For every partition size $n$ and parameter $z$,

$$\sum_{\lambda \vdash n}\prod_{h\in\mathcal{H}(\lambda)^{\diamond}}\left(\frac{h+z}{h}\right)=\sum_{\lambda\vdash n}\prod_{h\in\mathcal{H}(\lambda)}\left(\frac{h^2+z}{h^2}\right).$$

This conjecture refines the Nekrasov–Okounkov hook length formula. The paper proves it, so the identity is no longer open.