Random-information decay threshold for polynomially decaying ellipsoids

Let $1\le p\le\infty$ and let the semiaxes be $\sigma_j=j^{-\lambda}$ for $j\in\mathbb{N}$, with $\lambda>\frac{1}{s}=\left(\frac{1}{2}-\frac{1}{p}\right)_+$. Write $\mathcal{E}_{p,\sigma}^{m}$ for the corresponding finite-dimensional $\ell_p$-ellipsoid, $G_{n,m}$ for a random $n$-dimensional information subspace, and $\operatorname{rad}(\mathcal{E}_{p,\sigma}^{m},G_{n,m})$ and $\operatorname{rad}(\mathcal{E}_{p,\sigma}^{m},n)$ for the radii of random and optimal information, respectively. Let $p^*$ be the conjugate exponent of $p$. Random-information threshold conjecture.

$$\operatorname{decay}\bigl(\operatorname{rad}(\mathcal{E}_{p,\sigma}^{m},G_{n,m})\bigr)= \begin{cases} \operatorname{decay}\bigl(\operatorname{rad}(\mathcal{E}_{p,\sigma}^{m},n)\bigr),&\lambda>\frac{1}{p^*},\\ 0,&\lambda\le\frac{1}{p^*}. \end{cases}$$

The conjecture predicts a threshold at $\lambda=1/p^*$: above it, random information has the same decay rate as optimal information, while at or below it random information has no decay. The surrounding discussion establishes several regions of this picture, but the asserted threshold behavior remains open.