Is showing all trees have $\rho$-valuation not enough to prove Ringel's conjecture about trees decomposing odd complete graph?

Question

This might be a soft question, but I am trying to understand graceful labeling ($\beta$-valuation) and all the related stuff, and I have read Rosa's paper too.

I would like to know why most are focusing on showing that a tree always has $\beta$-valuation. Having that valuation implies having the $\rho$ one too, but are there trees having $\rho$-valuation that do not decompose their associated odd complete graph? I am trying to find papers on this but cannot find one. Maybe if we talk about graph in general, then maybe the answer is yes, but what happen if the graph is a tree?

Thanks in advance.