Deligne's generalized Thom–Sebastiani conjecture

Let $\Lambda$ be a noetherian torsion ring such that $m\Lambda=0$ for some integer $m$ relatively prime to $p$. Let $S_1$, $S_2$, and $S$ be henselian traits over $\operatorname{Spec} k$ of equal characteristic, all with residue field $k$, with closed points $s_1$, $s_2$, and $s$. Let $a:S_1\times_k S_2\to S$ be a $k$-morphism satisfying $a(s_1,s_2)=s$ and such that $a(\mathord\cdot,s_2):S_1\to S$ and $a(s_1,\mathord\cdot):S_2\to S$ are isomorphisms. For $i=1,2$, let $f_i:X_i\to S_i$ be of finite type, let $x_i$ be a $k$-rational point in the special fiber such that $f_i|_{X_i-x_i}$ is smooth, and let $K_i\in D^b_{\operatorname{ctf}}(X_i,\mathbb Z/\ell^m)$ with $\mathscr H^q(K_i)|_{X_i-x_i}$ locally constant for every $q$. Write $\eta_i$ for the generic point of $S_i$ and $j_i:\eta_i\hookrightarrow S_i$ for the canonical open immersion. Deligne's generalized Thom–Sebastiani conjecture. Under these conditions, $a\circ(f_1\times f_2)$ is locally acyclic relative to $K_1\boxtimes^L K_2$ outside $(x_1,x_2)$; consequently, $R\Phi_{X_1\times_kX_2}(K_1\boxtimes^L K_2)$ relative to $a\circ(f_1\times f_2)$ is supported at $(x_1,x_2)$. Moreover, there is a canonical isomorphism

$$R\Phi_{S_1\times_kS_2}\Big(j_{1!}\big((R\Phi_{X_1}(K_1))_{x_1}\big)\boxtimes^L j_{2!}\big((R\Phi_{X_2}(K_2))_{x_2}\big)\Big)_{(s_1,s_2)} \cong \Big(R\Phi_{X_1\times_kX_2}(K_1\boxtimes^L K_2)\Big)_{(x_1,x_2)},$$

where $R\Phi_{S_1\times_kS_2}$ is the vanishing-cycle functor for $a$, and $(R\Phi_{X_i}(K_i))_{x_i}$ are complexes of $\Lambda$-modules with $\operatorname{Gal}(\bar\eta_i/\eta_i)$-action, regarded as objects of $D^b_{\operatorname{ctf}}(\eta_i,\Lambda)$. This is a proposed extension of the Thom–Sebastiani theorem to general henselian traits and torsion coefficients; the supplied material attributes it to Deligne but gives no evidence that it has been proved or disproved.