The affine Temperley–Lieb quotient conjecture for induced subalgebras

Let $N_1,N_2$ be nonnegative integers with $N=N_1+N_2$. Write $\widehat{\mathsf{H}}_{N_i}$ for the affine Hecke algebras, $\mathsf{T}^a_{N_i}$ for the affine Temperley–Lieb algebras, and let

$$\mathsf{I}=\langle E_1,\omega\rangle\subset\widehat{\mathsf{H}}_{N}$$

be the defining ideal of the quotient $\widehat{\mathsf{H}}_{N}\twoheadrightarrow\mathsf{T}^a_N$. Let $\mathsf{I}_C=\mathsf{I}\cap(\widehat{\mathsf{H}}_{N_1}\otimes\widehat{\mathsf{H}}_{N_2})$.

Affine Temperley–Lieb quotient conjecture. There is an isomorphism of algebras

$$\widehat{\mathsf{H}}_{N_1}\otimes\widehat{\mathsf{H}}_{N_2}/\mathsf{I}_C\cong\mathsf{T}^a_{N_1}\otimes\mathsf{T}^a_{N_2}.$$

This identifies the quotient induced from the affine Hecke subalgebra with the tensor product of the corresponding affine Temperley–Lieb quotients, and would justify the affine Temperley–Lieb fusion construction used in the surrounding discussion. The source reports computations supporting the identification but does not provide a proof here.