Goncharov's polynomiality conjecture for cyclotomic motivic dimensions

For a positive integer $N$, define

$$d_w(N):=\operatorname{dim}{\cal Z}_w(S_N),\qquad d_{w,m}(N):=\operatorname{dim}\operatorname{Gr}^D_m{\cal Z}_w(S_N),$$

where ${\cal Z}_w(S_N)$ is the weight-$w$ component and $\operatorname{Gr}^D_m$ is the depth-$m$ graded component. Let $p$ range over prime numbers.

Goncharov's polynomiality conjecture. If $p$ is a prime, then $d_w(p)$ and $d_{w,m}(p)$ are polynomials in $p$.

The conjecture concerns uniform dimension formulas for cyclotomic motivic structures as the prime level varies. The supplied text does not state whether it has been proved or disproved.