Existence of ODD geodesics

Let $(M,\mathfrak{g})$ be an ODD Riemannian manifold, let $p\in M$, and suppose an ODD orthonormal frame $(E_i)_i$ is defined around $p$. Let $v\in T_pM$ be a tangent vector at $p$.

Existence of ODD geodesics conjecture. There exists a unique ODD geodesic $\gamma$ through $p$ in direction $v$.

The preceding theorem establishes this existence and uniqueness at general points of $M$ and at general points of maximal components of the degeneracy locus. The conjecture proposes the corresponding statement at arbitrary points where an ODD orthonormal frame is available; the general case is not proved in the supplied text.