Conjectured Gelfand and approximation number bound for Schatten embeddings

Let $S_p^N$ denote the Schatten class of $N\times N$ matrices, and let $a_n$ and $c_n$ denote the $n$th approximation and Gelfand numbers, respectively. For $2\leq p\leq q\leq\infty$ and $n,N\in\mathbb N$ satisfying

$$(1-c)N^2\leq n\leq N^2-cN^{1+2/q}+1,$$

The Gelfand–approximation conjecture. One has

$$c_n\big(\mathcal{S}_p^N\hookrightarrow\mathcal{S}_q^N\big)\leq a_n\big(\mathcal{S}_p^N\hookrightarrow\mathcal{S}_q^N\big)\lesssim_{p,q}\left(\sqrt{\frac{N^2-n+1}{N^2}}\right)^{\frac{1/p-1/q}{1/2-1/q}}.$$

The conjecture concerns the lower-left triangular region for Gelfand and approximation numbers; by duality and the cited lemma, it is equivalently related to corresponding Kolmogorov-number regions. The source presents this as unsettled.