The deficient path forest conjecture on impossibly burnable forests

For $m\in\mathbb{N}$ and $1\leq l\leq m^2$, let $B_m(l)$ be the least positive integer $t$ having the same parity as $l$ such that

$$l\leq\sum_{i=1}^t[2m-(2i-1)]=2mt-t^2.$$

Let $T=(l_1,l_2,\ldots,l_n)$ be an $n$-path forest of order $m^2$. Call $T$ deficient when it cannot be burned in $m$ rounds, and call it impossibly burnable when

$$\sum_{i=1}^n B_m(l_i)>m.$$

Deficient path forest conjecture. Let $n\geq4$. If $T$ is a deficient $n$-path forest with $l_1\geq L_{n-1}$, then $T$ is impossibly burnable. The conjecture is motivated by computational verification for small values of $n$, but its general validity remains open.