Letellier's conjecture on twisted mixed Hodge polynomials of partial character varieties

Let $\mathcal{M}_{\mu,\nu}$ be the partial resolution of the character variety defined in the source, and let $w$ be an $n$-cycle in the Weyl group relative to the third puncture. Let $IH_c^w(\mathcal{M}_{\mu,\nu};u,v)$ denote its $w$-twisted mixed Hodge polynomial, and let $c_{\mu,\nu}^{1^n}(q,t)$ be the corresponding structure coefficient. Letellier's conjecture. The coefficient satisfies

$$c_{\mu,\nu}^{1^n}(q,t)=t^{-\dim(\mathcal{M}_{\mu,\nu})/2}IH_c^w\left(\mathcal{M}_{\mu,\nu};\frac{1}{q},\sqrt{qt}\right).$$

The formula is presented as a consequence of Letellier's conjectural Weyl-group action formula for partial resolutions of character varieties; the source gives no evidence of a resolution.