GKS logarithm-removal conjecture for approximation numbers of Schatten embeddings

Let $S_p^N$ denote the Schatten class of $N\times N$ matrices with exponent $p$, and let $a_n$ be the $n$th approximation number of the embedding. For $1<p\leq q<\infty$ and $n,N\in\mathbb N$ satisfying

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

GKS's conjecture. The approximation numbers satisfy

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

This conjecture concerns the logarithmic factors in the upper bound for approximation numbers in the indicated region and is attributed in the paper to GKS (1987).