Proof of a conjecture of Fomichev and Karev

We prove a conjecture of Fomichev and Karev [{European J. Combin.} 127 (2025) 104160] by showing the equality of two graph invariants: $\phi$, defined via graph colorings, and $\psi$, derived from the $\mathfrak{s}\mathfrak{l}(2)$-weight system of its 2-dimensional irreducible representation.