The strengthened burning number conjecture for trees with a prescribed number of leaves

Let $T$ be a tree, and say that a graph is $m$-burnable when its burning number is at most $m$. Let $m>n\geq 2$, and suppose that $T$ has $n$ leaves and order at most

$$m^2+n-2.$$

The strengthened burning number conjecture. The tree $T$ is $m$-burnable.

This conjecture strengthens the usual burning number conjecture for trees by allowing trees of order larger than $m^2$. The source attributes it to earlier work, and notes that it holds for stars, paths, and spiders; the parser marks the stated conjecture as resolved.