The critical-time conjecture for stationary interfaces of the parabolic p-Laplace equation

Let $u$ solve the one-dimensional $p$-parabolic boundary-value problem

$$u_t- \Delta_p u=0$$

on $(-1,1)\times(0,T)$ with boundary values $u(-1,t)=0$, $u(1,t)=1$, and initial value $u(x,0)=x_+^{\frac{p}{p-2}}$. For the restriction to $(0,1)\times(0,T)$, let $c_p$ and $c_0$ be the constants from the preceding finite-speed-of-propagation estimate, and set $\hat t=(c_p c_0)^{p-2}$. Critical-time conjecture. The critical time $\hat t=(c_p c_0)^{p-2}$ is the largest time such that $u(0,t)=0$. This concerns whether the interface remains stationary exactly up to the time supplied by the estimate; the provided passage does not state whether the claim has been proved or disproved.