Bilinear relations for Fourier-transformed Nekrasov functions

Identify the root lattice of $A_{N-1}$ with

$$Q_{N-1}={(n_1,\ldots,n_N)\in\mathbb{Z}^{N}\mid\sum_i n_i=0}.$$

For $j\in\mathbb{Z}/N\mathbb{Z}$, define the Fourier-transformed Nekrasov functions by

$$\mathcal{T}^{N,k}_j(\vec{u},\vec{s};q\mid z)=\sum_{\vec{\Lambda}\in Q_{N-1}+\omega_j}s^\Lambda Z^{N,k}(\vec{u}q^{\vec{\Lambda}};q^{-1},q\mid z),$$

where $\omega_0=0$ and $\omega_j$ are the fundamental weights of $A_{N-1}$. Bilinear relations for Fourier-transformed Nekrasov functions. The functions $\mathcal{T}^{N,k}_j$ satisfy

$$\mathcal{T}^{N,k}_j(qz)\mathcal{T}^{N,k}_j(q^{-1}z)=\mathcal{T}^{N,k}_j(z)^2-z^{1/N}\mathcal{T}^{N,k}_{j+1}(q^{k/N}z)\mathcal{T}^{N,k}_{j-1}(q^{-k/N}z).$$

These relations are analogous to the blow-up equations for Nekrasov functions and connect the Fourier-transformed functions with bilinear Toda-type structures. They were conjectured previously and have been proven, so the statement is solved.