Tangency principle in backwards light cones for the relativistic heat equation

Let $U_T$ be the spacetime domain, let $U_{(x,t)}$ be the backwards light cone of $(x,t)\in U_T$, and let $\widetilde{Q}$ be the spatial operator in the relativistic heat equation. Tangency principle. Suppose $w,w'\in C^2_1(U_T)\cap C(\overline{U_T})$ satisfy

$$w_t-\widetilde{Q}w\leq w'_t-\widetilde{Q}w'$$

and $w\leq w'$ in $U_{(x,t)}$. If $w(x,t)=w'(x,t)$, then $w\equiv w'$ in $U_{(x,t)}$. This proposed strengthening of comparison and uniqueness would assert rigidity when two ordered sub- and supersolutions touch at the tip of a relativistic backwards light cone; the paper presents it as an expected principle requiring further techniques, so it remains open.