An origin centric ellipsoid is defined by any positive semi-definite $n$ by $n$ matrix $X$, by taking all vectors $v$ such that $v^tXv\leq1$. Call two origin centric ellipsoid *equivalent* if one can be obtained from the other by rotation of space.

Now suppose that $E$ and $E'$ are two equivalent centric ellipsoids that are both contained inside a third centric ellipsoid $F$. Is it always possible to continuously rotate $E$ inside $F$ until is gets to the position $E'$?

This question is a more difficult version of this one.