The vanishing integral conjecture

Let $\lambda\in P_{+}(2n)$ and let $a,b,q,t\in\mathbb C$ satisfy $|a|,|b|,|q|,|t|<1$. Define the normalized integral

$$I_{\lambda}(a,b;q,t,t^n):=\frac{1}{Z_n(a,b;q,t)}\int_{\mathbb T^n}P_{\lambda}(x_1^{\pm},\ldots,x_n^{\pm};q,t)\,\prod_{i=1}^n\frac{(x_i^{\pm2};q)_\infty}{(ax_i^{\pm2},bx_i^{\pm2};q^2)_\infty}\prod_{1\leqslant i<j\leqslant n}\frac{(x_i^{\pm}x_j^{\pm};q)_\infty}{(tx_i^{\pm}x_j^{\pm};q)_\infty}\,\operatorname{d}T(x).$$

Vanishing integral conjecture. The integral $I_{\lambda}(a,b;q,t,t^n)$ vanishes unless $\operatorname{2\text{-}core}(\lambda)=0$. Moreover, the two specializations satisfy

$$I_{\lambda}(q,t;q,t,T)=q^{n^{\mathrm e}(\lambda')-n^{\mathrm o}(\lambda')}t^{2\hat n^{\mathrm o}(\lambda)-2\hat n^{\mathrm e}(\lambda)}\frac{C^{0,\mathrm e}_{\lambda}(T^2;q,t)}{C^{0,\mathrm o}_{\lambda}(qT^2/t;q,t)}\frac{C^{-,\mathrm e}_{\lambda}(q;q,t)}{C^{-,\mathrm o}_{\lambda}(t;q,t)},$$

and

$$I_{\lambda}(1,qt;q,t,T)=\frac{u_{\lambda}(q,t)+v_{\lambda}(q,t)T}{1+T}\frac{C^{0,\mathrm e}_{\lambda}(T^2;q,t)}{C^{0,\mathrm o}_{\lambda}(qT^2/t;q,t)}\frac{C^{-,\mathrm e}_{\lambda}(q;q,t)}{C^{-,\mathrm o}_{\lambda}(t;q,t)},$$

with $u_{\lambda}$ and $v_{\lambda}$ as defined in the source. These identities give conjectural evaluations of special cases of the integral, while the general vanishing assertion is the key expected property.