They leave together
There are three levels. All of them are straight slides along the axes.
38 of the 65: a single piece slides straight out of the other seven. Whenever that happens, the whole cube comes apart, always in seven straight slides.
18 more: no single piece can move, but a pair of pieces can leave together. Once that pair is out, the rest separates. So 56 of the 65 can be taken apart, if more than one piece may move at a time.
9 cannot be separated at all. No subset of the eight — not a pair, not a half, not any of the 254 groups — can be translated away from the rest along an axis. They are 16, 34, 47, 48, 55, 59, 61, 64 and 65. They exist as mathematics, or as one fused object.
Stronger than the first level. In every one of the 38, all eight pieces can leave at the same time, each along a direction it is free to take, with no two meeting. That was checked continuously along the travel, not only at whole-cell steps.
The 38 where a single piece slides free are exactly the 38 whose piece has a way to sit with nothing hanging in the air. The other 27 are exactly the 27 whose piece always leaves one or two cubes unsupported, however you turn it. No exceptions among the 65.
Observed This is not a proof. A piece that wraps tightly enough that no single piece slides free is a piece that cannot be laid down flat. That is the intuition, not a theorem.
All of the counts above test straight translations along the three axes, of rigid groups, moving off to infinity. Rotating a piece on the way out, or sliding it on a curved or dog-leg path, was not tested. “Cannot be separated” means by straight axis motions. That is the usual first test. It is not a proof that no motion at all exists.