Trees T satisfying W(L3(T))= W(T)

Let G be a graph. Denote by Li(G) its i-iterated line graph and denote by W(G) its Wiener index. We find an infinite class of trees T satisfying W(L3(T)) = W(T), which disproves a conjecture of Dobrynin and Entringer [Electronic Notes in Discrete Math. 22 (2005) 469-475].