The motivic fundamental lemma

Let $\mathbb{K}$ be the motivic coefficient ring described in the source, let $(\mathfrak{g},\mathfrak{h})$ be a pair in Definition~, and let $r\in\mathbb{Q}$. For the varieties $U(\mathfrak{g},\mathfrak{h},r)_i$, coefficients $b(\mathfrak{g},\mathfrak{h},r)_i$, index sets $I(\mathfrak{g},\mathfrak{h},r)$, and polynomials $p(\mathfrak{g},\mathfrak{h},r)$ introduced earlier, the relevant classes lie in $\mathbb{K}$.

Motivic fundamental lemma. For every such pair and every $r\in\mathbb{Q}$, one has the identity in $\mathbb{K}$

$$p(\mathfrak{h},\mathfrak{h},r)(\mathbb{L})\sum_{i\in I(\mathfrak{g},\mathfrak{h},r)} b(\mathfrak{g},\mathfrak{h},r)_i[U(\mathfrak{g},\mathfrak{h},r)_i] =p(\mathfrak{g},\mathfrak{h},r)(\mathbb{L})\sum_{i\in I(\mathfrak{h},\mathfrak{h},r)} b(\mathfrak{h},\mathfrak{h},r)_i[U(\mathfrak{h},\mathfrak{h},r)_i].$$

This is the geometric or motivic form of the fundamental lemma: after cross-multiplying the denominators from the preceding variety formula, it predicts an identity of motivic classes. The source does not state a resolution status for this conjecture.