Balanced tree decomposition conjecture for complete bipartite graphs

A balanced tree $T_{n,n}$ has $n$ vertices in each partition class, hence $2n$ vertices in total. A graph decomposes $G$ if copies of the graph form a perfect packing of $G$.

Balanced tree decomposition conjecture. Any tree $T_{n,n}$ decomposes $K_{2n-1,2n-1}$.

This is presented as following from the Graham–Häggkvist conjecture for $n$-regular bipartite graphs. The source then asks whether a smaller, possibly asymmetric host could suffice; the displayed decomposition assertion itself remains open.