Alekseev's dense collision-orbit conjecture

Let $\mathcal{U}$ be a subset of the phase space, and call an initial condition a collision initial condition when its associated trajectory reaches a collision. A subset of the phase space is open if it contains a neighborhood of each of its points, and a set of initial conditions is dense in $\mathcal{U}$ if every nonempty open subset of $\mathcal{U}$ contains one of them.

Alekseev's dense collision-orbit conjecture. Is there an open subset $\mathcal{U}$ of the phase space such that for a dense subset of initial conditions the associated trajectories go to a collision?

Saari proved that collision orbits have zero measure, but the conjecture asks whether they can nevertheless be topologically rich. The source attributes this conjecture to Alekseev and notes that it may trace back to Siegel; its resolution is not stated here.