Non-local modulus of continuity on a bounded convex domain

Let $\Omega$ be a bounded convex domain in $\mathbb R^n$ with diameter $D$. Choose rectangular coordinates whose $z_1$-axis is aligned with the compact interval $I=[-D/2,D/2]$. For each $z_1\in I$, define the slice

$$\Omega_{z_1}=\{p\in\mathbb R^{n-1}:(z_1,p)\in\Omega\}.$$

Suppose $u:\Omega\times[0,\infty)\to\mathbb R$ satisfies the non-local regional heat equation, and let $\varphi$ be as in the one-dimensional modulus-of-continuity theorem. Assume that its odd extension $\tilde\varphi$ satisfies

$$\tilde\varphi_t(r,t)=\int_I\tilde\rho(w-r)\bigl(\tilde\varphi(w)-\tilde\varphi(r)\bigr)\,dw,\qquad r\in I,\ t>0,$$

where

$$\tilde\rho(z_1)=\int_{\Omega_{z_1}}\rho(z_1,p)\,dp.$$

Non-local modulus of continuity conjecture. Then $\varphi(\cdot,t)$ is a modulus of continuity for $u(\cdot,t)$ on $\Omega\times[0,\infty)$. This proposes that the higher-dimensional regional non-local heat flow on a bounded convex domain admits the same type of modulus-of-continuity control as the associated one-dimensional sliced equation. The statement is presented as a conjecture; no resolution is supplied in the source.