The curvature criterion for doubling geodesically convex subsets

Let $X$ be an $RCD(K,N)$ space, and let $\Omega$ be a geodesically convex open subset of $X$ with $\operatorname{m}(\partial\Omega)=0$. Let $\hat{\operatorname{m}}$ be the measure on $\operatorname{Dbl}(\overline\Omega)$ defined by

$$\hat{\operatorname{m}}(A)=\operatorname{m}(A\cap\Omega_0)+\operatorname{m}(A\cap\Omega_1)$$

for all $A\subset\operatorname{Dbl}(\overline\Omega)$, where $\Omega_0$ and $\Omega_1$ are the two copies of $\Omega$. Doubling curvature criterion. The measured doubled space $(\operatorname{Dbl}(\overline\Omega),\hat{\operatorname{m}})$ satisfies $RCD(K,N)$ if and only if

$$\Delta d_{\Omega^c}\leq-\sqrt{K(N-1)}\frac{\sin\left(\sqrt{\frac{K}{N-1}}d_{\partial M}\right)}{\cos\left(\sqrt{\frac{K}{N-1}}d_{\partial M}\right)}\quad\text{on }\Omega.$$

This is formulated as an if-and-only-if conjecture in the supplied candidate, although the surrounding text presents related Riemannian and $RCD$ doubling results; the notation $d_{\partial M}$ is not reconciled with the preceding $X,\Omega$ notation.