GL_n and SL_n affine Springer cyclicity conjecture

Let $cmathfrak g=\operatorname{Lie}(GL_n)$ or $cmathfrak g=\operatorname{Lie}(SL_n)$, and let $gamma\bin g(cO)$ be elliptic, regular semisimple, and topologically nilpotent. Let $[\operatorname{Sp}_gamma]$ be the fundamental class, with ${cC}[T^*T^\vee]^W$ acting by the stated convolution action. Affine Springer cyclicity conjecture.

$$H_*({\operatorname{Sp}}_\gamma)={cC}[T^*T^\vee]^W[\operatorname{Sp}_\gamma].$$

This is proposed to explain finite generation in the $GL_n$ and $SL_n$ elliptic cases; it remains open in the source.