Joyce–Song formula for motivic Milnor fibers

Let $E_1$ and $E_2$ be semi-Schur objects in the derived category. For a semi-Schur object $E$, let $L_E$ be the cyclic $L_\infty$-algebra $\operatorname{Ext}^*(E,E)$, let ${\mathcal S}_0(E)={\mathcal S}_{f,0}(L_E)$ be its motivic Milnor fiber, and let $f$ denote the corresponding formal potential. For a formal subscheme $\mathfrak Z$ of the formal scheme associated with $E$, write ${\mathcal S}_{\mathfrak Z}(\hat f)$ for the motivic Milnor fiber over $\mathfrak Z$. The pushforward of motivic vanishing cycles is denoted by

$$\int_{\mathfrak X_0}(-):{\mathcal M}^{\hat\mu}_{\mathfrak X_0}\to {\mathcal M}^{\hat\mu}_{\mathbb C}.$$

Joyce–Song formula. The following identities should hold:

$$1-\mathcal{S}_{((0,0))}(E_1\oplus E_2)=(1-\mathcal S_0(E_1))\cdot(1-\mathcal S_0(E_2)).$$

Moreover,

$$\begin{aligned} &\int_{F\in\mathbb P(\operatorname{Ext}^{1}(E_2,E_1))}(1-\mathcal S_0(F))-\int_{F\in\mathbb P(\operatorname{Ext}^{1}(E_1,E_2))}(1-\mathcal S_0(F))\\ &=\bigl([\mathbb P^{\dim\operatorname{Ext}^{1}(E_2,E_1)}]-[\mathbb P^{\dim\operatorname{Ext}^{1}(E_1,E_2)}]\bigr)\left(1-\mathcal S_{f|_{X_{E_1}\oplus X_{E_2}},0}\right). \end{aligned}$$

These identities are proposed as motivic analogues of the Joyce–Song formulas for extensions of semi-Schur objects. They extend the preceding Thom–Sebastiani formula for motivic Milnor fibers, but the source provides no resolution status.