Embedding balanced P3P3-designs into (balanced) P4P4-designs

Denote by P(v,k,1)P(v,k,1) a PkPk-design of order vv and block size k (every path has k   vertices and k-1k-1 edges). A P(v,k,1)P(v,k,1) having the property that every vertex meets the same number of paths [P. Hell, A. Rosa, Graph decompositions, handcuffed prisoners, and balanced PP-designs, Discrete Math. 2 (1972) 229–252] is called balanced. Denote by H(v,k,1)H(v,k,1) a balanced P(v,k,1)P(v,k,1).Let V⊂WV⊂W, |V|=v