Existence of a tetrad pack scale

It seems these exist for strong seven-limit vals, but not otherwise.
It makes me wonder if with the right definitions something along those
lines could be proven. Another curiosity is that for higher scale
divisions, the tetrad pack scale seems to acquire uniqueness, with
only two inverse scales qualifying. Putting these observations
together, it appears tetrad pack scales are rather rare, which is
another reason to find them interesting.

Such scales exist for the standard vals for 10, 12, 15, 19, 22, 27,
31, 41, 68, 72 and 99. Among the systems for which they don't exist I
found
9, 16, 17, 21, 25, 26, 37, 46, 53, and 58. Merely being a pretty good
7-limit system does not seem to be good enough.