Flatness implies smoothness for solutions of the porous medium equation

Let $\rho$ be a solution to the porous medium equation on $[-1,0] \times B_2(0)$. Assume that there is a nonempty open cone $C$ such that $\rho(t,x) \leq \rho(t,y)$ whenever $y-x\in C$. The flatness-implies-smoothness conjecture. Then $\rho^{m-1}$ is smooth up to the boundary of the support outside the initial support, and

$$\nabla \rho^{m-1} \ne 0$$

at the free boundary. This conjecture proposes that a one-sided monotonicity condition in an open cone prevents the focusing singularities that can obstruct free-boundary regularity for the porous medium equation. The supplied context explains that focusing can destroy regularity at intermediate times, while eventual smoothness is known for compactly supported solutions; the status of this local conjecture is not established in the supplied material.