On a conjecture about Wiener index in iterated line graphs of trees

Let GG be a graph. Denote by Li(G)Li(G) its ii-iterated line graph and denote by W(G)W(G) its Wiener index. Dobrynin and Melnikov conjectured that there exists no nontrivial tree TT and i≥3i≥3, such that W(Li(T))=W(T)W(Li(T))=W(T). We prove this conjecture for trees which are not homeomorphic to the claw K1,3K1,3 and HH, where HH is a tree consisting of 6 vertices, 2 of which have degree 3.