Infinite-product factorization of the universal K-matrix in affine type

Prove that, for quantum symmetric pairs of affine type, the universal basic K-matrix Y admits an infinite-product factorization analogous to the factorization of the (quasi-)R-matrix for quantum affine algebras and Yangians.

Background

In finite type, explicit factorizations of the universal K-matrix Y have been obtained by decomposing it according to reduced expressions of the longest element in the restricted Weyl group, mirroring known factorizations of quasi R-matrices. In affine type, however, explicit expressions are lacking, and the presence of imaginary roots complicates the structure.

This conjecture seeks an affine analogue of the classical infinite-product factorizations of R-matrices (as in Khoro­shkin–Tolstoy) for the universal K-matrix Y, thereby providing a structured and computable form of Y in the affine setting.