Exact-spectrum conjecture for interval colorings of K_{1,m,n}

Let $K_{1,m,n}$ be the complete tripartite graph with part sizes $1,m,n$, and let an interval $t$-coloring mean an interval edge-coloring using the colors $1,\ldots,t$. Exact-spectrum conjecture. Graph $K_{1,m,n}$ has an interval $t$-coloring if and only if

$$t=m+n\quad\text{and}\quad \gcd(m+1,n+1)=1.$$

This is posed as a future-work conjecture about the exact number of colors. The source proves non-colorability when the gcd exceeds $1$, but does not resolve the full assertion.