Nekrasov–Piazzalunga's explicit formula for the rank-equal Nekrasov genus

Let $X$ be the affine four-space occurring in the definition of the Nekrasov genus, let $E=\bigoplus_{i=1}^{N}\mathcal{O}_X\langle y_i\rangle$ and $V=\bigoplus_{i=1}^{N}\mathcal{O}_X\langle v_i\rangle$, and define

$$[x]=x^{\frac12}-x^{-\frac12}.$$

Set $s=\prod_{i=1}^{N}y_iv_i$. Nekrasov–Piazzalunga's conjecture. There exists a choice of signs $o(\mathcal{L})$ such that

$${\mathcal{N}}_X(E,V;q)=\operatorname{Exp}\left(\frac{[t_1t_2][t_2t_3][t_1t_3]}{[t_1][t_2][t_3][t_4]}\frac{[s]}{[s^{\frac12}q][s^{-\frac12}q]}\right).$$

The paper uses this conjectural formula to deduce admissibility of the Nekrasov genus in the rank-equal case; its resolution is not established in the source.