The strict support-dimension conjecture for Serre relations

Let $K=\mathbb{C}$ and let $\Pi=\Pi(C,D)$. Define the support of a constructible function $f\colon \operatorname{nil}_E(\Pi,d)\to\mathbb{C}$ by

$$\operatorname{supp}(f)=\{M\in\operatorname{nil}_E(\Pi,d)\mid f(M)\not=0\}.$$

Let $\widetilde{{\mathcal M}}(\Pi)$ be the convolution algebra of constructible functions and let $\mathcal I$ be the ideal generated by the Serre relations. Strict support-dimension conjecture. For every $0\not=f\in\widetilde{{\mathcal M}}(\Pi)_d\cap\mathcal I$,

$$\dim\operatorname{supp}(f)<\dim G(d)-q_{DC}(d/D).$$

The preceding dimension estimate gives only the non-strict inequality for arbitrary constructible functions; this conjecture asserts strictness for every nonzero element of the Serre-relation ideal and is the key condition needed to control the quotient by those relations.