Prime-dimensional stabilizer 3-design conjecture

Let $d$ be prime, let $D=d^n$, and let $\mathrm{Stab}(n,d)$ denote the ensemble of stabilizer states. The source considers its shadow norm $\|\mathrm{Stab}(n,d)\|_{\mathrm{sh}}$. Prime-dimensional stabilizer 3-design conjecture.

$$\|\mathrm{Stab}(n,d)\|_{\mathrm{sh}}=\frac{D+1}{D+d}\left(2d-1-\frac{d}{D}\right).$$

For $d=2$, the lower bound is known to be attained because stabilizer states form a 3-design; the conjecture proposes equality for every prime local dimension. The source gives no resolution for odd primes.