Reduction conjecture for interval colorings of complete tripartite graphs

Let $K_{k,m,n}$ be the complete tripartite graph with part sizes $k,m,n$, where $k\leq m\leq n$ and $n>k+m$. Reduction conjecture.

$$K_{k,m,n}\text{ is interval colorable}\quad\Longleftrightarrow\quad K_{k,m,n-k-m}\text{ is interval colorable}.$$

This is proposed as a generalization of the $K_{1,m,n}$ colorability problem; the source gives no resolution of the reduction in this range.