Infinite-volume limit of Bethe Ansatz orthogonality for Grothendieck polynomials

Prove that, in the specialization b_1=\cdots=b_{n-1}=-\beta, the limit as M\to+\infty of the finite Bethe Ansatz orthogonality relation for the symmetric Grothendieck polynomials G_\lambda and \overline{G}_\mu yields the torus-integral orthogonality relation established for these polynomials.

Background

The paper establishes a torus scalar-product orthogonality relation for inhomogeneous symmetric Grothendieck polynomials. It also cites a finite-dimensional orthogonality relation for the homogeneous specialization b_1=\cdots=b_{n-1}=-\beta, formulated as a sum over roots of Bethe Ansatz equations and involving a parameter M.

The relationship between these two orthogonality formulas is left as a conjectural limiting statement: the cited discrete relation is expected to converge to the torus-integral relation when M tends to infinity.

References

Above the weight $w_M$ depends on the positive integer $M$ and letting $M\to +\infty$ it is natural to conjecture that one should recover eq:orthogonality Grothendieck polys, for the particular case $b_1=\cdots=b_{n-1}=-\beta$.

— Orthogonality of spin $q$-Whittaker polynomials  (2502.00478 - Mucciconi, 1 Feb 2025) in Section 5, subsection “Inhomogeneous symmetric Grothendieck polynomials,” Remark following Theorem 5.2