Finite-order vanishing Laplace Hopf-lemma conjecture

Let $u$ and $v$ satisfy the hypotheses of the preceding strong Laplace Hopf-lemma conjecture, including $u_t(0,0)=0$. Finite-order vanishing conjecture. If, in addition, $u(t,0)$ and $v(t,0)$ vanish at $t=0$ to finite order, then

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

The source calls this the weaker of two conjectures for a Laplace-operator variation of the Hopf Lemma; it is presented without a resolution.