Minimax error bound for the column sparse principal subspace case

Let $q\in[0,1]$, and let $\hat{\mathcal{S}}$ estimate the principal subspace $\mathcal{S}$ under the same conditions as in the column sparse upper-bound corollary, with $R_q$ replaced by $dR_q$. Write $\|\sin\Theta(\hat{\mathcal{S}},\mathcal{S})\|_F^2$ for the squared Frobenius subspace error. Minimax error bound for the column sparse case. There exists an estimator $\hat{\mathcal{S}}$ such that, with high probability,

$$\|\sin\Theta(\hat{\mathcal{S}},\mathcal{S})\|_F^2\leq c d R_q\left(\frac{\sigma^2(1+\log p)}{n}\right)^{1-q/2},$$

and consequently the optimal minimax lower and upper bounds satisfy

$$\|\sin\Theta(\hat{\mathcal{S}},\mathcal{S})\|_F^2\asymp d R_q\left(\frac{\sigma^2\log p}{n}\right)^{1-q/2}.$$

This would close the gap between the existing column-sparse upper and lower bounds when $d$ is larger than the logarithmic regime; the source provides no resolution of the conjecture.