Conjecture on computing Whittaker integrals with boundary polynomials

Fix a root system $\Phi$ and a long word $\underline{i}$. Let ${\bf m}\in\mathcal{B}(-\infty)$ be an $\underline{i}$-Lusztig datum, and let $\lambda$ be a dominant coweight. Let $g_i(t_\alpha,w_\alpha)$ be the Laurent polynomials constructed from the diagonal specialization procedure, and let $C^{\underline{i}}({\bf m})$, $f(u)$, and $\psi$ denote the corresponding integration domain, integrand factor, and additive character.

Boundary-polynomial conjecture.

$$I_\lambda({\bf m})=\int_{C^{\underline{i}}({\bf m})}f(u)\psi\left(\sum_{i\in I}\varpi^{\lambda_{i^\ast}}g_i(t_\alpha,w_\alpha)\right)\,du.$$

Equivalently, the Laurent polynomials $h_i(t_\alpha,w_\alpha)$ in the general formula may be replaced by $g_i(t_\alpha,w_\alpha)$ for all $\underline{i}$-Lusztig data. The claim is presented as an expected extension beyond cases where the two polynomial constructions agree and is not established in the supplied text.