Monotonicity conjecture for the moment of inertia along three-body brake orbits

Assume a brake orbit segment in the three-body problem, beginning at brake time and ending at its first syzygy. Let $I(t)$ denote the total moment of inertia of the three bodies. Moment-of-inertia monotonicity conjecture. Along every such brake orbit segment, including the first syzygy time, one has

$$\dot I(t)<0.$$

If true, this would imply that no brake orbit other than Lagrange's has a first intersection with the syzygy plane orthogonal, because orthogonality requires $\dot I=0$ at the intersection. The source presents this as a conjectural non-existence statement; no resolution is supplied here.