Nondegeneracy and compatibility of coisotropic intersections

Let $X$ be a derived Artin stack, and let $L_1\longrightarrow X$ and $L_2\longrightarrow X$ be morphisms equipped with $n$-shifted coisotropic structures. Write $\mathrm{Cois}^{nd}(f_i,n)$ for the spaces of nondegenerate coisotropic structures and $\mathrm{Pois}^{nd}(X,n)$ and $\mathrm{Pois}^{nd}(L_1\times_X L_2,n-1)$ for the spaces of nondegenerate shifted Poisson structures. Also write $\mathrm{Lagr}(f_i,n)$ and $\mathrm{Symp}(X,n)$ and $\mathrm{Symp}(L_1\times_X L_2,n-1)$ for the corresponding spaces of shifted Lagrangian and symplectic structures. Coisotropic intersection property. (1) If $L_1\longrightarrow X$ and $L_2\longrightarrow X$ carry nondegenerate $n$-shifted coisotropic structures, then the induced $(n-1)$-shifted Poisson structure on $L_1\times_X L_2$ is nondegenerate. (2) The diagram of spaces

\xymatrix{ \mathrm{Cois}^{nd}(f_1, n)\times_{\mathrm{Pois}^{nd}(X, n)} \mathrm{Cois}^{nd}(f_2, n) \ar[r] \ar^{\sim}[d] & \mathrm{Pois}^{nd}(L_1\times_X L_2, n-1) \ar^{\sim}[d] \\ \mathrm{Lagr}(f_1, n)\times_{\mathrm{Symp}(X, n)} \mathrm{Lagr}(f_2, n) \ar[r] & \mathrm{Symp}(L_1\times_X L_2, n-1) }

is commutative. These properties express the compatibility between the intersection constructions for shifted coisotropic and shifted Lagrangian structures; the parser supplies no evidence resolving the statement, so its status remains open.