Saunderson–Chandrasekaran–Parrilo–Willsky ellipsoid-fitting conjecture

An origin-symmetric ellipsoid in ${\varmathbb R}^d$ is specified by an equation

$$\langle x,Qx\rangle=1,$$

for a positive semidefinite matrix $Q\succeq 0$. Let $z_1,\ldots,z_n$ be independent random points with

$$z_1,\ldots,z_n\sim\mathcal N\left(0,\frac{1}{d}\cdot I_d\right).$$

Saunderson–Chandrasekaran–Parrilo–Willsky conjecture. For arbitrarily small $\varepsilon>0$, if $n<(\frac14-\varepsilon)\cdot d^2$, then with high probability over $z_1,\ldots,z_n$, there exists an ellipsoid passing through all the points.

The problem asks how many random Gaussian points can lie on a common ellipsoid and is motivated by recovering a subspace from noisy measurements through semidefinite programming. The conjecture was based on experimental results; the supplied source gives no resolution status, so it is recorded as open.