The intermediate-range minimum/maximum degree conjecture

Let $k$ be a positive integer and let $\alpha\in[0,\frac13)$. Intermediate-range conjecture. Every graph with minimum degree at least $(1+\alpha)\frac k2$ and maximum degree at least $2(1-\alpha)k$ contains every tree with $k$ edges. This interpolates between the two degree conjectures above. The source gives partial results but no general proof.