The maximal rank of a string group generated by involutions for alternating groups

A string group generated by involutions, or SGGI, is a pair $\mathrm{\Gamma }=(G,S)$, where $G$ is a group and $S=\{{\rho }_0,\dots ,{\rho }_{r-1}\}$ is an ordered set of involutions generating $G$ and satisfying the commuting property: $$\mathrm{\forall }i,j\in \{0,\dots ,r-1\},\mathrm{\mid }i-j\mathrm{\mid }\mathrm{\ne }1\Rightarrow ({\rho }_i{\rho }_j{)}^2=1.$$ When $S$ is an independent set, the rank of $\mathrm{\Gamma }$ is the cardinality of $S$. We determine an upper bound for the rank of an SGGI over the alternating group of degree $n$. Our bound is tight when $n\equiv 0,1,4(\mathrm{m}\mathrm{o}\mathrm{d}5)$.