Alekseev's conjecture on the measure of oscillatory motions

Let $OM$ denote the set of oscillatory motions in the restricted $3$-body problem, namely the set of points whose forward and backward limit sets under the map $P_\mu$ both contain the parabolic fixed point $O$, excluding the set $P$ of parabolic motions:

$$OM=\{z=(x,y)\in\mathbb{R}^2\colon O\in\omega_P(z)\cap\alpha_P(z)\}\setminus P.$$

Alekseev's conjecture. The set $OM$ has Lebesgue measure zero.

Oscillatory motions are a distinctive phenomenon of the restricted $3$-body problem, and their abundance is related to the structure of the homoclinic class of the parabolic fixed point. The supplied source presents this as Alekseev's conjecture and gives no evidence of a resolution.