Rosenberg–Stolz conjecture for products with the plane

Let $Y$ be a closed manifold of dimension $(n-2)\neq4$ which does not admit a metric of positive scalar curvature.

Rosenberg–Stolz conjecture. $Y\times\mathbb{R}^2$ does not admit a complete metric with uniformly positive scalar curvature.

This is the codimension-two counterpart of the Rosenberg–Stolz conjecture for products with the real line. The paper's abstract states that it is solved up to dimension $7$ for orientable manifolds.