$\mathrm{SL}_2$-factorisations of Chevalley groups

Recently Liebeck, Nikolov, and Shalev noticed that finite Chevalley groups admit fundamental $\mathrm{SL}_2$-factorizations of length $5N$, where $N$ is the number of positive roots. From a recent paper by Smolensky, Sury, and Vavilov it follows that elementary Chevalley groups over rings of stable rank 1 admit such factorizations of length $4N$. In the present paper, we establish two further improvements of these results. Over any field the bound here can be improved to $3N$. On the other hand, for $\mathrm{SL}(n,R)$, over a Bezout ring $R$, we further improve the bound to $2N=n^2-n$.