A Finiteness Theorem for Universal $m$-Gonal Forms with coefficients $1$ or $2$

In 2022, Kim \cite{k} proved a finiteness theorem for a restricted class of universal generalized $m$-gonal forms; namely, a generalized $m$-gonal form $f$ with coefficients $1$ or $2$ is universal if $m\ge10$ and $f$ represents $1$, $m-4$ and $m-2$. In this paper, we prove a similar finiteness theorem for universal $m$-gonal forms. If $m$ is even, $m\ge10$ and an $m$-gonal form $f$ with coefficients $1$ or $2$ represents $2m-1$ and $4m-2$, then $f$ is universal, and if $m$ is odd, $m\ge7$ and $f$ represents either $2m-1$ and $2m-2$ or $2m-2$ and $5m-4$, then $f$ is universal.