The seven pairs are [6,8], [8,12], [12,24], [15,40], [16,48], [20,120], [21,168]. The list is definite, but the conjecture is unproved. The conjecture asserts that

"Sum_{ak+1 square} p(n-k) == 1 mod 2 if and only if bn+1 is a square" holds if and only if [a,b] is one of these seven pairs.

Here p(n) is the number of partitions of n, A000041.

REFERENCES

Ballantine, Cristina, and Mircea Merca. "Parity of sums of partition numbers and squares in arithmetic progressions." The Ramanujan Journal 44.3 (2017): 617-630.