The linear upper-bound conjecture for path-forest burning thresholds

For each $n\geq4$, let $L_n$ denote the threshold parameter for $n$-path forests introduced in the paper. Linear upper-bound conjecture.

$$L_n\leq12n\quad\text{for all }n\geq2.$$

If true, this would give a good asymptotic approximation to the values of $L_n$ and provide an alternative route to estimating the path-forest burning thresholds. Its general validity is open.