The sectional-curvature characterization of Alexandrov spaces among RCD spaces

Let $({\rm M},{\sf d})$ be a complete and separable metric space. An $n$-dimensional Alexandrov space has curvature bounded from below by $k\in\mathbb{R}$ in the Alexandrov sense. Suppose moreover that $({\rm M},{\sf d},\mathcal H^n)$ is an ${\sf RCD}(k(n-1),n)$ space, that $\operatorname{supp}(\mathcal H^n)={\rm M}$, and that $\bm{\mathcal R}$ denotes the distributional Riemann curvature operator on test vector fields and test functions.

Sectional-curvature characterization conjecture. The following conditions are equivalent: $({\rm M},{\sf d})$ is an $n$-dimensional Alexandrov space of curvature bounded from below by $k$, and

$$\bm{\mathcal R}(X,Y,Y,X)(f)\geq k\int f|X\wedge Y|^2\,\mathrm d\mathfrak m$$

for every $f\in{\rm Test}({\rm M})$ with $f\geq0$ and every $X,Y\in{\rm TestV}({\rm M})$.

This would characterize the Alexandrov condition through the lower bound of the abstract sectional curvature on finite-dimensional ${\sf RCD}$ spaces. The paper presents it as an equivalence between the Alexandrov curvature bound and the stated ${\sf RCD}$, support, and curvature-operator conditions; no resolution is given.