The base conjecture for graded retracts of monomial quotients

Let $S=k[x_1,\ldots,x_n]$ be the standard graded polynomial ring, let $S_+=(x_1,\ldots,x_n)$, and let $I$ be a monomial ideal satisfying

$$I\subseteq (S_+)^2.$$

A base of a graded algebra retract of $S/I$ is the distinguished set of variables arising in the paper's construction of such a retract.

Base conjecture. Every graded algebra retract of $S/I$ has a base.

The conjecture is motivated by the paper's classification results for graded algebra retracts of affine monoid rings and Stanley–Reisner rings. The supplied text does not state a resolution or a known special case, so its status remains open.