The dimension conjecture for T-prime ideals

Let $\Bbbk$ be a field of characteristic $p>0$, and let a T-prime ideal mean an ideal of $\Bbbk S_\infty$ satisfying the T-prime condition introduced in the source. A dimension of such an ideal refers to the dimension notion associated with the T-system formalism in the paper.

Dimension conjecture. Every T-prime ideal admits a dimension.

This is stated as a consequence predicted by the preceding conjecture for positive characteristic. The source gives no proof or resolution of this assertion.