  We prove that for $c>0$ a sufficiently small universal constant that a random
set of $c d^2/\log^4(d)$ independent Gaussian random points in $\mathbb{R}^d$
lie on a common ellipsoid with high probability. This nearly establishes a
conjecture of~\cite{SaundersonCPW12}, within logarithmic factors. The latter
conjecture has attracted significant attention over the past decade, due to its
connections to machine learning and sum-of-squares lower bounds for certain
statistical problems.
