I am trying to solve the following question, that I guess (hope?) has been solved before but I couldn't find any reference.

Let $S$ be a set of $d$ unit vectors in a $d$-dimensional Euclidean space such that for any pair of vectors in $S$ their distance is the range $[\sqrt 2 -O(1/d), \sqrt 2 + O(1/d)]$.

Then, one can find a set $Q$ of $d$ pairwise orthogonal unit vectors such that there is a 1-to-1 mapping between vectors in $S$ and $Q$ such that each point of $S$ is mapped to a point in $Q$ at distance at most $O(1/d)$.

Note: If the range was $\sqrt 2\pm c/\sqrt d$ for some large constant $c$, that would not necessarily be true, here the fact that we have a $1/d$ additive term is key. But I am not sure how to use it...