Sharp bounds for successive minimum spanning tree weights

For each $k\geqslant1$, let $\gamma_k$ be the limiting constant for the weight of the $k$th successive minimum spanning tree. Sharp weight bounds conjecture. For every $k\geqslant1$,

$$2k-1\leqslant\gamma_k\leqslant2k.$$

These bounds would sharpen the proved estimates $2k-2\sqrt{k}<\gamma_k<2k+2\sqrt{k}$ and are motivated by taking increments of the known bounds for cumulative weights; they remain unproved.