The $\mathfrak{sl}(2)$ weight system at $c=-3/32$ extends to a graph 4-invariant

We prove unconditionally the conjecture of Fomichev, Karev, Pavutnitskiy and Usanov that the specialization of the $\mathfrak{sl}(2)$ weight system at $c=-3/32$ extends to a $4$-invariant of graphs. The extension is $\xi(G)=(-32)^{-n}S_G(-2)$, where $S_G(N)$ counts with signs the assignments of $2$-element subsets of $\{1,\dots,N\}$ to the $n$ vertices of $G$.