Eventual residue-class bias for unrestricted partitions and related partitions

Let $d_n(a,b;m):=p_n(a,b,m;0,1)$ denote the number of partitions of $n$ for which the relevant residue-class statistic indexed by $a$ exceeds that indexed by $b$. For positive integers $a,b,m$ satisfying $1\le a<b\le m$ and $m\ge 3$, with $(a,b,m)\ne(1,2,3)$, define the comparison threshold $N_{a,b,m}$ by the eventual inequality below.

Eventual bias conjecture. There exists a constant $N_{a,b,m}>0$ such that

$$d_n(a,b;m)\ge d_n(b,a;m),\qquad\text{for all }n\ge N_{a,b,m}.$$

Moreover,

$$N_{a,m-a,m}=0,\qquad\text{for all }1\le a<m/2,$$

except that $N_{2,3,5}=45$, $N_{2,4,6}=5$, $N_{3,4,7}=8$, and $N_{4,5,9}=9$. The conjecture proposes an eventual ordering of the two residue-class counts; the paper motivates it through generating functions and extensive computer experiments, while the general assertion remains open.