Finite-time convergence conjecture for sandpiles in a silo

Let $\Omega$ be the silo domain, let $J$ be the set of final points of the transport rays, and let $u_\phi$ denote the maximal profile. Assume that $f\in L^1_+(\Omega)$ satisfies

$$\sup_{y\in J}\int_0^{1} \frac{|B_{\rho}|}{\int_{B_{\rho}(y)\cap\Omega} f(x)\, dx}\, d\rho < +\infty.$$

Finite-time convergence conjecture. Then $u(t,\cdot)$ converges to $u_\phi$ in finite time.

The condition requires sufficient source mass near every final point of the transport rays. The preceding radial examples show that failure of the analogous integrability condition can prevent finite-time convergence; whether the stated sufficient condition holds in the full silo model remains open.