The converse Alexandrov-to-distributional sectional curvature conjecture for $C^1$ metrics

Let $(M,g)$) be a Riemannian manifold with $gcin C^1$ such that $(M,d_g)$ is an Alexandrov space with curvature bounded from below (respectively, above) by $k cin cmathbb{R}$. Distributional sectional curvature is the curvature notion defined by distributional inequalities for the curvature tensor of a low-regularity metric.

Converse curvature conjecture. The distributional sectional curvature of $(M,g)$ is bounded from below (respectively, above) by $k$.

This conjecture asks for a converse to the paper's result that, for $C^1$ metrics, distributional sectional-curvature bounds imply the corresponding Alexandrov curvature bounds. Its status is not established in the supplied text.