Algebra retracts and Stanley-Reisner rings

In a paper from 2002, Bruns and Gubeladze conjectured that graded algebra retracts of polytopal algebras over a field k are again polytopal algebras. Motivated by this conjecture, we prove that graded algebra retracts of Stanley-Reisner rings over a field k are again Stanley-Reisner rings. Extending this result further, we give partial evidence for a conjecture saying that monomial quotients of standard graded polynomial rings over k descend along graded algebra retracts.