Summary: Let \(P^*_n\) denote the graph obtained by joining a new vertex to every vertex of a path on \(n\) vertices. Let \(U_{i,j}(n)\) denote the set of all connected graphs obtained from \(P^*_i\cup P^*_j\) by connecting the four vertices of degree 2 by two paths of lengths \(s\) \((\geq 0)\) and \(t\) \((\geq 1)\) such that \(s+t= n-i-j\) is a constant. N.-Z. Li and E. G. Whitehead jun. [Discrete Math. 122, No. 1-3, 365-372 (1993; Zbl 0787.05040)] conjectured that \(U_{3,4}(n)\) forms a chromatic equivalence class by itself. In this note we prove the conjecture in the affirmative.