The square power partition function inequality conjecture

Let $p^k(n)$ denote the number of partitions of $n$ into perfect $k$-th powers. In particular, $p^2(n)$ is the square power partition function, and let $a,b$ be positive integers.

Square power partition function inequality conjecture. The inequality

$$p^2(a)p^2(b)\geq p^2(a+b)$$

has equality and failure exactly in the instances listed in the source's table: equality for $a=1$, $b=1,2,4,5,6,9,10,13,14,18,22$; $a=2$, $b=4,5,9,13$; $a=3$, $b=4$; $a=4$, $b=5,6,7$; $a=5$, $b=5,6,8,11,15$; $a=6$, $b=8,10,12,14$; and $a=7$, $b=8,9,12,13,19$, with the listed complementary instances of failure.

The claim refines the eventual multiplicative inequality for restricted partition functions by identifying its exceptional small-index behavior. The source gives no resolution beyond presenting these instances as a conjectural pattern.