For which values of $n$ can we tile some rectangle with one copy of each free simply-connected $n$-omino (that is, each polyomino with $n$ squares that has no holes)?

It appears that it is possible for $n=1$ (trivial), $n=2$ (trivial), $n=5$ (see here), and $n=7$ (see here); and impossible for $n=3$ (trivial), $n=4$ (by a parity argument), and $n=6$ (by a parity argument). Is it known to be possible or impossible for any $n\geq 8$?