Bowen-type saddle-connection conjecture for vector fields

Let $M$ be a manifold, let $P_x^T$ denote the linear Poincare flow along the trajectory of a vector field $G$, and suppose that for an open, dense, full-Lebesgue-measure subset of $M$ one has

$$\liminf_{T\to\infty}\frac{1}{T}\ln\|(P_x^T)^{-1}\|<0$$

and

$$\liminf_{T\to\infty}\frac{1}{T}\ln\|P_x^T\|<0.$$

Bowen-type saddle-connection conjecture. Then $G$ exhibits saddle connections similar to the saddle connections in Example 1.

The claim is motivated by the Bowen example and predicts a characteristic saddle-connection configuration under simultaneous forward and inverse contraction on a large set; it remains unresolved in the supplied text.