Stable-center single-cycle generation conjecture

Let $q$ be a prime power, let $\Phi$ be the set of monic irreducible polynomials over $\mathbb F_q$ other than $t$, and let $\mathscr{G}(q)$ be the stable graded algebra with single cycle class sums $K_{(r)_f}$ indexed by $r\geq 1$ and $f\in\Phi$.

Stable-center generation conjecture. The stable center

$$\mathbb Q\otimes_{\mathbb Z}\mathscr{G}(q)$$

is the polynomial algebra generated by the single cycle class sums $K_{(r)_f}$, for all $r\geq 1$ and $f\in\Phi$.

This conjecture gives an explicit polynomial description of the stable center in terms of elementary cycle classes. The source provides no resolution or further evidence sufficient to determine its status.