Deficiency bounds for complete graphs of power-of-two order

Let $q\in\mathbb{Z}_+$, let $K_{2^{q+1}+1}$ be the complete graph on $2^{q+1}+1$ vertices, and let $w_{def}(G)$ and $W_{def}(G)$ denote the minimum and maximum, respectively, of the number of colors in a proper edge-coloring of $G$ having minimum deficiency.

Complete-graph deficiency conjecture. For any $q\in\mathbb{Z}_+$,

$$w_{def}\left(K_{2^{q+1}+1}\right)=W_{def}\left(K_{2^{q+1}+1}\right)=3\cdot2^q.$$

The conjecture is motivated by the inability to find a proper edge-coloring of these complete graphs with more than $3\cdot2^q$ colors having minimum deficiency. It remains open in the supplied source.