BAE from gauge origami on a toric Calabi–Yau three-fold times a line

Let $Z=X\times\mathbb{C}$, where $X$ is a toric Calabi–Yau three-fold, and consider the associated gauge origami system. In the Nekrasov–Shatashvili limit $q_{4}\to 1$, let $\mathfrak{q}_{i}$ be the gauge parameter, let $\mathrm{c}(x)$ denote the color of a variable $x$, and let $\varphi^{\mathrm{c}(x)\Rightarrow\mathrm{c}(x')}$ be the corresponding quiver structure function. The product below is over the relevant set of variables.

Gauge-origami Bethe ansatz conjecture. The Bethe ansatz equations are

$$1=-\mathfrak{q}_{i}\prod_{x'}\varphi^{\mathrm{c}(x)\Rightarrow\mathrm{c}(x')}(x'/x).$$

After adding flavors specified by additional polynomials $a_i(x)$ and $d_i(x)$, the equations become

$$1=-\mathfrak{q}_{i}\frac{a_i(x)}{d_i(x)}\prod_{x'}\varphi^{\mathrm{c}(x)\Rightarrow\mathrm{c}(x')}(x'/x).$$

Here $a_i(x)$ and $d_i(x)$ specify representations of the quiver quantum toroidal algebra. The claim is proposed as a generalization based on the BPS $qq$-character and quantum-toroidal-algebra description; a detailed analysis is deferred to future work.