Finite generation of the universal evaluation kernels

Prove that every kernel \(\ker \Phi_k\) of the universal \(k\)-fold evaluation homomorphism \(\Phi_k:\mathcal U(\mathfrak g[s,s^{-1}])\to(\mathcal U(\mathfrak g)[t,t^{-1}])^{\otimes k}\) is finitely generated as a two-sided ideal of \(\mathcal U(\mathfrak g[s,s^{-1}])\).

Background

The paper proves that the ideals $\ker\Phi_k$ form a cofinal family in the lattice of nonzero two-sided ideals of the enveloping algebra of the loop algebra $\mathfrak g[s,s^{-1}]$: every nonzero two-sided ideal contains $\ker\Phi_k$ for all sufficiently large $k$. It also establishes that these kernels are generated in a controlled way by discriminant elements only up to containment, rather than by an explicit finite generating set.

The authors state that they believe each $\ker\Phi_k$ is finitely generated, but do not prove this. Establishing finite generation would strengthen the structural description of the two-sided ideal lattice and clarify the role of the universal evaluation homomorphisms.