Hou–Jagadeesan maximal partition-rank product conjecture modulo 2

For a partition $\lambda=(\lambda_1,\ldots,\lambda_k)$, define

$$N(r,t;\lambda):=\prod_{j=1}^{k}N(r,t;\lambda_j).$$

Let $P(n)$ be the set of partitions of $n$, and define

$$\operatorname{maxN}(r,t;n):=\max\bigl\{N(r,t;\lambda):\lambda\in P(n)\bigr\}.$$

Hou–Jagadeesan's maximal-value conjecture. The following are true:

1. If $n\geq 5$, then

$$\operatorname{maxN}(0,2;n)= \begin{cases} 3^{n/3} & n\equiv 0\pmod{3},\\ 11\cdot 3^{(n-7)/3} & n\equiv 1\pmod{3},\\ 5\cdot 3^{(n-5)/3} & n\equiv 2\pmod{3}, \end{cases}$$

and the maximum is attained at the unique partitions $(3,3,\ldots,3)$, $(3,3,\ldots,7)$, and $(3,3,\ldots,5)$ in the respective congruence classes modulo $3$. If $n\geq 8$, then

$$\operatorname{maxN}(1,2;n)= \begin{cases} 2^{n/2} & n\equiv 0\pmod{2},\\ 12\cdot 2^{(n-9)/2} & n\equiv 1\pmod{2}, \end{cases}$$

and the maximum is attained by $(2,2,\ldots,2)$ when $n$ is even and $(2,2,\ldots,9)$ when $n$ is odd, up to any number of substitutions $(2,2)\to(4)$ and $(2,2,2)\to(6)$. The source presents this as a conjecture of Hou and Jagadeesan, without supplying a resolution in the provided text.