Groups whose same-order types are arithmetic progressions

For any group $G$, define $g\sim h$ if $g,h\in G$ have the same order. The set of sizes of the equivalent classes with respect to this relation is called the same-order type of $G$. In this short note we prove that there is no finite group whose same-order type is an arithmetic progression of length $4$. This answered an open problem posed by Lazorec and Tˇarnˇauceanu