Existence and directional uniqueness of ODD integral curves

Let $(M,\mathfrak{g})$ be an ODD Riemannian manifold, let $p\in M$, and let $X$ be an ODD vector field without zeros. An ODD regular integral curve is a regular integral curve in the ODD sense.

Existence and directional uniqueness conjecture. There exists $\epsilon>0$ and at least one ODD regular integral curve $\gamma\colon(-\epsilon,\epsilon)\to M$ with $\gamma(0)=p$. If $\gamma$ and $\mu$ are two integral curves through $p$, and $\epsilon$ is chosen small enough that $\gamma$ and $\mu$ are regular away from $p$, then

$$\lim_{t\to 0}\frac{\dot{\gamma}(t)}{|\dot{\gamma}(t)|_{\mathfrak{g}}}\neq\lim_{t\to 0}\frac{\dot{\mu}(t)}{|\dot{\mu}(t)|_{\mathfrak{g}}}.$$

The claim extends existence of integral curves beyond general points of maximal components, while allowing non-uniqueness in the usual parametrized sense; uniqueness is asserted through the limiting normalized directions. The surrounding discussion presents this as a conjectural extension, and no resolution is given.