Davison–Meinhardt's conjecture on motivic nearby fibers of weighted homogeneous functions

Let $Y$ be a smooth $k$-variety with the trivial ${\mathbb G}_{m,k}$-action, and let ${\mathbb G}_{m,k}$ act on ${\mathbb A}^n_k$ with weights $w_1,\ldots,w_n>0$. Let

$$f:{\mathbb A}^n_k\times_k Y\to {\mathbb A}^1_k$$

be a ${\mathbb G}_{m,k}$-equivariant function, where ${\mathbb G}_{m,k}$ acts on ${\mathbb A}^1_k$ with weight $d>0$. Davison–Meinhardt's conjecture. The motivic nearby fiber of $f$ equals $[f^{-1}(1)]$ in $\mathcal{M}^{\widehat{\mu}}_k$, where the $\widehat{\mu}$-action on $f^{-1}(1)$ factors through $\mu_d(k)$ and is given by

$$\mu_d(k)\times f^{-1}(1)\to f^{-1}(1): (\zeta,(x_1,\ldots,x_n,y))\mapsto (\zeta^{w_1}x_1,\ldots,\zeta^{w_n}x_n,y).$$

The conjecture is stated as a motivic nearby-fiber formula for weighted homogeneous functions. The source explains that the general case is proved in the paper; earlier work established special cases, including equal weights and, further, weight $d=1$.