Conjectured Kolmogorov-number lower bound for quasi-Banach Schatten embeddings

Let $S_p^N$ and $S_q^N$ be Schatten classes of $N\times N$ matrices, and let $d_n$ denote the $n$th Kolmogorov number of the embedding. For $0<p\leq\infty$, $0<q\leq1$, $q\leq p$, and $n,N\in\mathbb N$ satisfying $1\leq n\leq N^2$, the quasi-Banach Kolmogorov-number conjecture. One has

$$d_n\big(\mathcal{S}_p^N\hookrightarrow\mathcal{S}_q^N\big)\gtrsim_{p,q}\max\left\{1,\frac{N^2-n+1}{N}\right\}^{1/q-1/p}.$$

The conjecture concerns Kolmogorov numbers when the codomain is a quasi-Banach Schatten space; the source gives no resolution.