First, the strong Stone–Weierstrass conjecture for postliminal ${\rm JB}^*$-triples is proved. This is combined with the recent classification of ${\rm JB}^*$-triples $A$ for which $\partial_e(A^*_1)$ is ${\rm weak}^*$ dense in $A^*_1$,and the weak Stone–Weierstrass theorem for ${\rm JB}^*$-triples is obtained. If $B$ is a ${\rm JB}^*$-subtriple of a ${\rm JB}^*$-triple $A$ such that $B$ separates $\overline{\partial_e(A^*_1)}\cup\{0\}$, then $B=A$.