Ramsey equivalence of $K_n$ and $K_n+K_{n-1}$ for multiple colours

In 2015 Bloom and Liebenau proved that $K_n$ and $K_n+K_{n-1}$ possess the same $2$-Ramsey graphs for all $n\ge 3$ (with a single exception for $n=3$). In the following we give a simple proof that $K_n$ and $K_n+K_{n-1}$ possess the same $r$-Ramsey graphs for all $n,r\ge 3$.