The two-thirds minimum/maximum degree conjecture

Let $G$ be a graph, and let $k$ be a positive integer. Two-thirds conjecture. If $\delta(G)\ge\left\lfloor\frac{2k}{3}\right\rfloor$ and $\Delta(G)\ge k$, then $G$ contains every tree with $k$ edges. The conjecture seeks to replace the Erdős–Sós average-degree condition by simultaneous minimum- and maximum-degree conditions. The source records the spanning case and other partial results, but not a general resolution.