Spin q-Whittaker polynomial orthogonality conjecture

Let $N$ be fixed, let $\boldsymbol{z}=(z_1,cdots,z_N)in eqmathbb{T}^N$, and let $\mathbb{F}_lambda$ and $\mathbb{F}_mu$ be the spin $q$-Whittaker polynomials indexed by signatures $lambda$ and $mu$. Define

$$m_{q,s}^N(z_1,\dots,z_N)=\frac{1}{N!}\prod_{1le ine jle N}\frac{(s^2,z_i/z_j;q)_\infty}{(-sz_i,-s/z_i;q)_\infty}\prod_{i=1}^Nfrac{1}{2pimathrm{i}z_i}.$$

Spin q-Whittaker orthogonality conjecture. For all signatures $\lambda,mu$,

$$\int_{\mathbb{T}^N}\mathbb{F}_{\lambda}(z_1,\dots,z_N)\mathbb{F}_{mu}(1/z_1,\dots,1/z_N)m_{q,s}^N(z_1,\dots,z_N),dz_1\cdots dz_N=c_{\lambda}\mathbf{1}_{\lambda=mu},$$

where

$$c_{\lambda}=\prod_{i=1}^{N-1}\frac{(s^2;q)_\infty}{(q;q)_\infty}\frac{(q;q)_{\lambda_i-\lambda_{i+1}}}{(s^2;q)_{\lambda_i-\lambda_{i+1}}}.$$

This is the conjectural spin deformation of the torus orthogonality of $q$-Whittaker polynomials. The corresponding spin Hall--Littlewood polynomials have related spatial and biorthogonality identities, while the displayed orthogonality remains conjectural.