Strong Laplace Hopf-lemma conjecture for ordered graph functions

Let $u\ge v$ be in $C^\infty(\overline\Omega)$, where $\Omega$ is the domain specified earlier in the source. Assume $u>0$, $v>0$, and $u_t>0$ in $\Omega$, with $u(0,y)=0$ for $|y|<1$. Whenever $u(t,y)=v(s,y)$ for $0<t\le s<1$, assume

$$\Delta u(t,y)\le\Delta v(s,y).$$

Strong Laplace Hopf-lemma conjecture. If, in addition, $u_t(0,0)=0$, then

$$u\equiv v\quad\text{in }\Omega.$$

This is the stronger of two conjectures attributed to Li and Nirenberg; the source presents variations of the Hopf Lemma as open problems and gives no resolution for this statement.