$(2,3)$-generated groups with small element orders

A periodic group is called an $OC_n$-group if the set of its element orders consists of all natural numbers from $1$ to some natural $n$. W. Shi posed the question whether every $OC_n$-group is locally finite. Until now, the case $n=8$ remains open. Here we prove that if a group is generated by an involution and an element of order $3$, and its element orders do not exceed $8$, then it is finite. Thereby we obtain an affirmative answer to Shi's question for $n=8$ for $(2,3)$-generated subgroups.