The unbounded Littlewood identities

Let $x=(x_1,x_2,\ldots)$ be an infinite set of variables, and let $\kappa^{(1)}_{\lambda}(q,t)$ and $\kappa^{(2)}_{\lambda}(q,t)$ be the limiting coefficients defined in the source. Unbounded Littlewood identities. One has

$$\sum_{\lambda}\kappa^{(1)}_{\lambda}(q,t)P_{\lambda}(x;q,t)=\prod_{i\geqslant1}\frac{(tx_i^2;q^2)_\infty}{(x_i^2;q^2)_\infty}\prod_{i<j}\frac{(tx_ix_j;q)_\infty}{(x_ix_j;q)_\infty},$$

and

$$\sum_{\lambda}\kappa^{(2)}_{\lambda}(q,t)P_{\lambda}(x;q,t)=\prod_{i\geqslant1}\frac{(qtx_i^2;q^2)_\infty}{(qx_i^2;q^2)_\infty}\prod_{i<j}\frac{(tx_ix_j;q)_\infty}{(x_ix_j;q)_\infty}.$$

These identities are presented as the large-$m,n$ limit of the bounded identities and are conjectural in the paper.