Motivic McKay correspondence conjecture for the group scheme $\alpha_p$

Let $k$ be a perfect field of characteristic $p>0$, let $G=\alpha_p$, and let $V$ be a finite-dimensional representation of $G$ with quotient $V/G$. Let $H=\mathbb{Z}/p\mathbb{Z}$ and let $W$ be the $H$-representation corresponding to $V$. Write $D_{\mathbf d}$ for the invariant determined by the representation, and let $M_{\mathrm{st}}$ and $M_{\mathrm{st},o}$ denote the motivic stringy invariants, with associated moduli spaces $\Delta_G$ and motivic measure $\mu_G$. Motivic McKay correspondence conjecture. If $D_{\mathbf d}\ge2$, then the following equalities hold in $\hat{\mathcal{M}}'\cup\{\infty\}$:

$$M_{\mathrm{st}}(V/G)_{o}=\int_{\Delta_G}\mathbb{L}^{-\mathrm{sht}_{\mathbf d}}\,d\mu_G=M_{\mathrm{st}}(W/H)_{o},$$ $$M_{\mathrm{st}}(V/G)=\int_{\Delta_G}\mathbb{L}^{-\mathrm{sht}_{\mathbf d}'}\,d\mu_G=M_{\mathrm{st}}(W/H).$$

This is the paper's proposed motivic McKay correspondence for $\alpha_p$ and predicts equality with the corresponding invariant for the cyclic group of order $p$. The paper presents examples as supporting evidence, but no resolution of the conjecture is given.