Invariance of the number of minimal generators by degree for S-subalgebras

Let $K\langle X_d\rangle$ be the free associative algebra on $X_d$, equipped with the operation $\circ$, and let $(F,\circ)$ be an $S$-subalgebra. A minimal generating system is a generating system minimal with respect to inclusion, and its elements are graded by degree. Generator-count conjecture. Every two minimal generating systems of $(F,\circ)$ have the same number of elements of each given degree. This would extend the analogous invariance known for homogeneous free generating systems; the source does not provide a resolution, so the conjecture remains open.