Bruns–Gubeladze conjecture on graded algebra retracts of polytopal algebras

Let $k$ be a field, let $A$ be a polytopal algebra over $k$, and let $B$ be a graded algebra retract of $A$. Bruns–Gubeladze conjecture. $B$ is again a polytopal algebra over $k$. This conjecture is part of the program on polytopal extensions of classical results in linear algebra. Except for some special cases, including algebra retracts of dimension $2$, it remains wide open in general.