Matrix product factorization conjecture for complete graphs of order $4n+1$

Let $K_{4n+1}$ be a complete graph on $4n+1$ vertices, where

$$4n=m_1\cdot m_2$$

and both $m_1$ and $m_2$ are even. A graphical pair $((G,g),(H,h))$ without loops consists here of graphs $G$ and $H$ that are $m_1$-regular and $m_2$-regular, respectively. Matrix product factorization conjecture. There is such a graphical pair satisfying

$$K_{4n+1}=G_g\ast{}_h H.$$

This asserts the existence of a matrix-product factorization of the complete graph under the stated evenness and regularity conditions. The supplied passage does not indicate whether the conjecture has been resolved.