The conjectural recursive upper bound for Schur numbers

For $n\ge 2$, let $S(n)$ denote the $n$th Schur number. The recursive Schur-number upper-bound conjecture.

$$S(n)\le n\bigl(S(n-1)+1\bigr)\qquad\text{for all }n\ge 2.$$

This conjecture follows from the preceding conjecture on $L(n)$ together with the paper's upper-bound theorem, and it is verified in the source for $2\le n\le 5$.