The transversality conjecture for essential periodic-orbit neighborhoods

Let $G$ be an ordered vector field, let $H$ be the fixed invariant one-dimensional set, and let $T_1,\ldots,T_k$ be the periodic orbits in $Ess(G)$. Choose tubular neighborhoods $T'_1,\ldots,T'_k$ such that $G$ is transverse to each boundary torus $\partial T'_i$. Transversality conjecture. As $G$ is deformed relative to $H$ back to $F$, $F$ remains transverse to the possibly knotted tori $\partial T'_i$, for $i=1,\ldots,k$. If true, this would preserve the Conley index inside each $T'_i$ and constrain the invariant sets that can replace the essential periodic orbits; no proof is supplied.