The constant-degree-tree version of the $2k$–$\frac{k}{2}$ conjecture

Let $k,\Delta\in\mathbb{N}$, and let $G$ be a graph. Write $\delta(G)$ and $\Delta(G)$ for the minimum and maximum degrees of $G$, respectively. Constant-degree-tree conjecture. If

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

then $G$ contains every tree $T$ with $k$ edges satisfying $\Delta(T)\leq\Delta$.

This is a bounded-maximum-degree refinement of the $2k$–$\frac{k}{2}$ conjecture, weakening the required host maximum degree for trees of maximum degree at most $\Delta$. The source gives an approximate result supporting it, but the exact conjecture remains open.