Finiteness of interpolating (q,t)-characters χ_{i,a}

Prove that for every node i in the Dynkin diagram and every a in C*, the interpolating (q,t)-character χ_{i,a} obtained via mixed specialization has finitely many monomial terms; equivalently, show that the algorithm computing χ_{i,a} terminates.

Background

Interpolating (q,t)-characters were introduced to link the Grothendieck rings of finite-dimensional representations of U_q(ĝ) and its Langlands dual via a deformation, and a mixed specialization is used here to define χ_{i,a}.

In non simply-laced types, these χ_{i,a} are not a priori known to be finite sums, unlike the simply-laced situation where they coincide with q-characters of fundamental modules.

Finiteness would ensure that χ_{i,a} provide computable building blocks for constructing χ(Z) and for formulating truncation-parametrization statements.