Strong maximum and minimum principle in backwards light cones

Let $U_T$ be the spacetime domain, let $U_{(x,t)}$ denote the backwards light cone

$$U_{(x,t)}=\bigl\{(\xi,\tau)\in U_T:\lvert x-\xi\rvert\leq t-\tau\bigr\},$$

and let $\widetilde{Q}$ be the spatial operator in the relativistic heat equation. Strong maximum and minimum principle. If $w\in C^2_1(U_T)\cap C(\overline{U_T})$ satisfies $w_t-\widetilde{Q}w\leq0$ and attains an interior maximum $M$ at a point $(x,t)\in U_T$, then $w\equiv M$ in the backwards light cone $U_{(x,t)}$. Likewise, if $w$ satisfies $w_t-\widetilde{Q}w\geq0$ and attains an interior minimum $m$ at $(x,t)\in U_T$, then $w\equiv m$ in $U_{(x,t)}$. This would extend the classical strong maximum and minimum principle from parabolic cylinders to the causally relevant regions of the relativistic heat equation; it is presented as a possible strengthening requiring new techniques and remains open.