The interpolating degree-condition conjecture for tree embeddings

Let $k\in\mathbb{N}$, let $0\leq\alpha\leq \frac{1}{2}$, and let $G$ be a graph. Write $\delta(G)$ and $\Delta(G)$ for the minimum and maximum degrees of $G$, respectively. Interpolating degree-condition conjecture. If

$$\delta(G)\geq (1+\alpha)\frac{k}{2}\qquad\text{and}\qquad \Delta(G)\geq 2(1-\alpha)k,$$

then $G$ contains every tree with $k$ edges.

This conjecture interpolates between the degree requirements in the $2k$–$\frac{k}{2}$ conjecture as $\alpha$ varies, trading a stronger minimum-degree condition for a weaker maximum-degree condition. The source cites related forthcoming work and leaves this general statement open.