The integral identity conjecture for motivic Milnor fibers

Let $f$ be an element of $\kappa[[x,y,z]]$ invariant under the $\kappa^{\times}$-action of weight $(1,-1,0)$, with $f(0,0,0)=0$. Let $\mathfrak X$ be the formal neighborhood of $\mathbb A_{\kappa}^{d_1}$ in $\mathbb A_{\kappa}^d$, with structural morphism $\hat f$ induced by $f(x,y,z)$, and let $\mathfrak Z$ be the formal neighborhood of $0$ in $\mathbb A_{\kappa}^{d_3}$, with structural morphism $\hat f_{\mathfrak Z}$ induced by $f(0,0,z)$. Integral identity conjecture. The identity

$$\int_{\mathbf x\in\mathbb A_{\kappa}^{d_1}}\mathcal S_{\hat f,\mathbf x}=[\mathbb A_{\kappa}^1]^{d_1}\mathcal S_{\hat f_{\mathfrak Z},0}$$

should hold in $\mathscr M_{\kappa}^{\hat\mu}$. This identity is the motivic form of the integral identity underlying the construction of motivic Donaldson–Thomas invariants; the source discusses its motivic and $\ell$-adic versions, but provides no resolution of the conjecture here.