The Planted Affine Planes SoS conjecture for nearly quadratic sample size

Let $n$ be the ambient dimension, let $m$ be the number of sampled vectors, and let SoS refer to the sum-of-squares hierarchy. The Planted Affine Planes problem is the hypothesis-testing problem considered in the source, with theorem

$$giving its current SoS bound. **Planted Affine Planes SoS conjecture.** Theorem$$

holds with the bound on the number of sampled vectors loosened to

$$m \leq n^{2-\varepsilon}.$$

This conjecture predicts that the Planted Affine Planes problem remains difficult for SoS with nearly $n^2$ sampled vectors; the source presents it as an open problem, and the stated upper-bound threshold is motivated by the behavior of the degree-bounded pseudoexpectation on the constant polynomial.