Nonexistence of nontrivial prime gradings on matrix algebras

Let $K$ be an algebraically closed field of characteristic zero, let $G$ be a finite nontrivial abelian group, and let $n>3$. A $G$-grading $\Gamma$ on $M_n(K)$ is called nontrivial when it is not the trivial grading, and the grading satisfies the primeness property for graded central polynomials when the product of two graded central polynomials in disjoint variable sets being graded central forces each factor to be graded central.

Nonexistence conjecture. No nontrivial $G$-grading $\Gamma$ on $M_n(K)$ satisfies the primeness property for graded central polynomials.

The paper proves the corresponding nonexistence result for matrix orders $2$ and $3$, and presents the statement for $n>3$ as the natural extension. Its resolution remains open in the supplied text.