The pure 5d Nekrasov partition-function relation at special parameters

Let $\mathcal{Z}_{\mathrm{inst}}(i,i,i,iq^{\pm1/2},u;q^{-1},q|z^{1/2})$ denote the five-dimensional Nekrasov instanton partition function with the displayed special values of $v_i$, and let $\mathcal{Z}_{\mathrm{inst}}(u;q^{-1},q^2|z)$ denote the pure five-dimensional Nekrasov instanton partition function with $\epsilon_2=-2\epsilon_1$. Nekrasov partition-function relation. These partition functions satisfy

$$(-qz^{1/2};q,q)^2_{\infty}\mathcal{Z}_{\mathrm{inst}}(i,i,i,iq^{\pm1/2},u;q^{-1},q|z^{1/2})=\mathcal{Z}_{\mathrm{inst}}(u;q^{-1},q^2|z).$$

The relation is presented as a consequence expected from the tau-function correspondence and was checked computationally through order $z^5$ in the surrounding text; no general proof is supplied.