Can there be an energetically unbounded three-body orbit where escape is impossible due to conservation of angular momentum?

Question

This question evolved from a discussion below this answer which explains (among other things) that the total energy of a system offers insight as to the possibility of one (or all) members "escaping".

The total energy would be the sum of the kinetic and potential energies

$$E = \sum_{i=1}^{3}\frac{1}{2}m_i v_i^2 - \sum_{i=1}^{3} \sum_{j>i}^{3} \frac{m_i m_j}{r_{ij}}.$$

Can there be some three body orbit that is energetically unbounded ($E>0$) but where it is still impossible for any of the objects to escape due to do conservation of angular momentum?

Possibly helpful: Equations of motion for the n-body problem

notes:

1. I'm not asking if there exist orbits that are closed and periodic where escape is impossible for that reason.
2. I haven't written an expression for angular momentum because there is flexibility about which point it is calculated.

Batominovski's Clarification on the Bounty (as noted by Angela Pretorius in a comment). The energy should be measured with respect to the center-of-mass frame of the system. That is, the condition $$\sum_{i=1}^3m_iv_i=0$$ is enforced.

^†Based on comments here and my suspicion I've corrected $i \ne j$ to $i > j$ for the potential energy term to avoid double-counting.

Answer 1

The answer is no... conservation of angular momentum, by itself, can't be used to prove boundedness of a 3-body system with positive total energy (in the frame where the center of mass is stationary at the origin). For sufficiently large $t$, all escaping bodies (there must be at least 2) will have essentially fixed velocities ${\bf v}_i$ and linearly evolving positions ${\bf x}_i + t {\bf v}_i$. The total angular momentum is $\sum_i \left({\bf x}_i + t{\bf v}_i\right) \times m_i{\bf v}_i = \sum_i m_i{\bf x}_i \times{\bf v}_i$, also a constant. But note that the angular momentum can be changed to any value without changing the total energy, the total momentum, or the center of mass, by adding appropriate offsets to the ${\bf x}_i$. (Keeping the center of mass fixed imposes one vector constraint on these offsets; since at least two bodies are escaping, there is at least one vector degree of freedom remaining.)

In short, conservation of angular momentum doesn't help you because each "escape scenario" belongs to an equivalence class of scenarios (with the same total energy and momentum) that differ only in their angular momenta.