Hou–Jagadeesan convexity conjecture for partition ranks modulo 2

Let $N(r,2;n)$ denote the number of partitions of $n$ whose rank is congruent to $r$ modulo $2$, for $r\in\{0,1\}$. Hou–Jagadeesan's conjecture. If $r=0$ (respectively, $r=1$), then

$$N(r,2;a)N(r,2;b)>N(r,2;a+b)$$

for all $a,b\geq 11$ (respectively, $a,b\geq 12$). The paper states that this conjecture is proved by its main theorem, establishing the claimed convexity bound modulo $2$ after earlier work on the analogous modulus-$3$ result.