  We conduct quasi-Monte Carlo numerical integrations in two very high (80 and
79)-dimensional domains -- the parameter spaces of rank-9 and rank-8
qutrit-qutrit (9 x 9) density matrices. We, then, estimate the ratio of the
probability -- in terms of the Hilbert-Schmidt metric -- that a generic rank-9
density matrix has a positive partial transpose (PPT) to the probability that a
generic rank-8 density matrix has a PPT (a precondition to
separability/nonentanglement). Close examination of the numerical results
generated -- despite certain large fluctuations -- indicates that the true
ratio may, in fact, be 2. Our earlier investigation (eprint quant-ph/0410238)
also yielded estimates close to 2 of the comparable ratios for qubit-qubit and
qubit-qutrit pairs (the only two cases where the PPT condition fully implies
separability). Therefore, it merits conjecturing (as Zyczkowski was the first
to do) that such Hilbert-Schmidt (rank-NM/rank-(NM-1)) PPT probability ratios
are 2 for all NM-dimensional quantum systems.
