Boundary curvature-integral conjecture for Alexandrov spaces

Let $(X,d_X)$ be an $n$-dimensional Alexandrov space with curvature bounded below by $-1$, and let $p\in\partial X$. Suppose that

$$B_2(p)\cap\partial X\setminus\mathcal{S}^{n-3}(X)$$

is a smooth Riemannian manifold with metric $g_{\partial X}$ induced by $d_X$. Write $scal_{\partial X}$ for its intrinsic scalar curvature.

Boundary curvature-integral conjecture. One has

$$\int_{B_1(p)\cap\partial X} scal_{\partial X}\,\operatorname d {vol}_{g_{\partial X}}\le C(n).$$

The conjecture is motivated by the expected Alexandrov-space structure of the boundary and by examples showing that a global Alexandrov condition is necessary for such a bound; the source gives no resolution of this estimate.