Open quipus with the same Wiener index as their quadratic line graph

An open quipu is a tree constructed by attaching a pendant path to every internal vertex of a path. We show that the graph equation W(L2(T))=W(T) has infinitely many non-homeomorphic solutions among open quipus. Here W(G) and L(G) denote the Wiener index and the line graph of G respectively. This gives a positive answer to the 2004 problem of Dobrynin and Mel'nikov on the existence of solutions with arbitrarily large number of arbitrarily long pendant paths, and disproves the 2014 conjecture of Knor and Å krekovski.