Uniqueness for solutions with different vanishing orders

Let Equation

have a solution $u(t)$ satisfying the initial conditions

with $m\geq n$. A second solution $v(t)$ is assumed to satisfy

$$v(0)=v'(0)=\cdots=v^{(l-1)}(0)=0,\qquad v^{(l)}(0)=b\neq 0,$$

where $l<n$.

The uniqueness conjecture. Such a solution $u(t)$ cannot possess another solution $v(t)$ satisfying these initial conditions.

This asks whether the uniqueness result remains valid when one solution has vanishing order below the order $n$ of the differential equation. The preceding theorem establishes uniqueness when both vanishing orders are at least $n$, while the case $l<n$ is left open.