The Planted Boolean Vector SoS conjecture for large random-subspace dimension

Let $n$ be the ambient dimension, let $p$ be the dimension of a random subspace, and let SoS refer to the sum-of-squares hierarchy. The Planted Boolean Vector problem is the dual problem considered in the source, with theorem

$$giving its current SoS bound. **Planted Boolean Vector SoS conjecture.** Theorem$$

holds with the bound on the dimension $p$ of a random subspace loosened to

$$p \geq n^{1/2+\varepsilon}.$$

This is presented as the dual analogue of the Planted Affine Planes conjecture and predicts SoS hardness for random subspaces of dimension at least $n^{1/2+\varepsilon}$. The source states it in the open-problems discussion, with no resolution supplied.