What methods exist to speed up testing of this Gilbreath-like conjecture?

What methods exist to speed up testing of this Gilbreath-like conjecture?

Gilbreath's conjecture is well-known for being particularly easy to
explain without any mathematical formalities.
Write out the first n primes,
2, 3, 5, 7, 11, 13, 17
take their differences,
1, 2, 2, 4, 2, 4
then the absolute values of their differences again and again.
1, 0, 2, 2, 2
1, 2, 0, 0
1, 2, 0
1, 2
1
The first term is always one, or so the conjecture says.
Note: the same has been conjectured of practical numbers (1, 2, 4, 6, 8,
12, 16, 18, 20, 24, 28)
The problem I'm interested in is the same operation performed on the
highly abundant numbers, which are those having a greater sum of divisors
than that of any smaller number. The highly abundant numbers begin:
1, 2, 3, 4, 6, 8, 10, 12, 16, 18, 20, 24, 30, 36, 42, 48, 60, 72, 84, 90,
96, 108, 120, 144
After taking the differences several times we start to notice a pattern in
the first few values of each iteration.
1, 1, 1, 2, 2, 2, 2, 4, 2, 2, 4, 6
0, 0, 1, 0, 0, 0, 2, 2, 0, 2, 2
0, 1, 1, 0, 0, 2, 0, 2, 2, 0
1, 0, 1, 0, 2, 2, 2, 0, 2
1, 1, 1, 2, 0, 0, 2, 2
0, 0, 1, 2, 0, 2, 0
0, 1, 1, 2, 2, 2
1, 0, 1, 0, 0
1, 1, 1, 0
0, 0, 1
The first three values cycle through {1, 1, 1}, {0, 0, 1}, {0, 1, 1}, {1,
0, 1} repeatedly. However, this pattern does not hold in general. I can't
remember exactly which iteration is the first to break the pattern, but
some parts within the pattern have been consistent through all iterated
differences for the first 246 highly abundant numbers. Namely, those rows
starting with {0, 0, 1} and {1, 0, 1} early on - equivalently those for
which the index of the iteration is even - continue to have {0, x, 1} and
{1, y, 1} through the first 246 iterations. This is as far as I've tested
so far. I suspect that some if not all of this pattern remains consistent
indefinitely.
My question: Can we apply a variant of the method used by Andrew Odlyzko
for the prime Gilbreath conjecture, described here, to test whether all or
part of this pattern holds for all highly abundant numbers up to a given
value without having to calculate every difference?
If you feel this post would benefit from the use of more formal
definitions feel free to submit an edit.