Local finiteness conjecture for the synthetic curvature measure

Let $X\in\text{Alex}^n(-1)$, and let

$$\mu_X={\operatorname R}\,\operatorname d\mathcal{H}^n+\mathcal{K}\,\operatorname d\mathcal{H}^{n-1}+\theta\,\operatorname d\mathcal{H}^{n-2}$$

be the synthetic curvature measure, where ${\operatorname R}$, $\mathcal{K}$, and $\theta$ are the corresponding density functions. Local finiteness conjecture. One has

$$\int_{B_1}\mu_X\le C(n).$$

This asks whether the synthetic curvature measure is locally finite. The source presents it as an open question and does not provide a resolution.