The quotient-motive conjecture for irreducible torus-knot representations

Let $n,m$ be coprime positive integers, and let

$$\Gamma_{n,m}=\langle x,y\mid x^n=y^m\rangle.$$

For a positive integer $r$, write

$$\mathcal{R}^{\mathrm{irr}}_r=\{(A,B)\in \operatorname{GL}_r(\mathbb{C})^2:A^n=B^m\}_{\mathrm{irr}}$$

for the irreducible representation locus, and let $\mathcal{M}^{\mathrm{irr}}_r=\mathcal{R}^{\mathrm{irr}}_r/\operatorname{PGL}_r(\mathbb{C})$. Quotient-motive conjecture. For every positive integer $r$,

$$[\mathcal{M}^{\mathrm{irr}}_r]=\frac{[\mathcal{R}^{\mathrm{irr}}_r]}{[\operatorname{PGL}_r(\mathbb{C})]}=\frac{[\mathcal{R}^{\mathrm{irr}}_r]}{q^{r-1}(q^r-1)(q^r-q)\cdots(q^r-q^{r-2})}$$

in the localization of $K_0(\operatorname{Var}_{\mathbb{C}})$ by $q$ and $q^i-1$ for $i=0,\ldots,r$.

This conjecture expresses the motive of the moduli space of irreducible representations as the quotient of the representation-locus motive by the projective linear group motive. The supplied text says that it is proved for $r\leq 3$, while the assertion for arbitrary positive $r$ remains open in the supplied context.