Berkovich–Uncu's eventual partition inequality for $C^{*}_{L,s}$ and $D_{L,s}$

For positive integers $L$ and $s$ with $L\geq s+1$, let $C^{*}_{L,s}$ be the set of partitions whose smallest part is $s$, whose parts are at most $L+s$, and in which $L$ does not occur. Let $D_{L,s}$ be the set of nonempty partitions with parts in $\{s+1,\ldots,L+s\}$. Berkovich and Uncu's conjecture. For positive integers $L\geq 3$ and $s$, there exists an $M$ depending only on $s$ such that

$$\left|\{\pi\in C^{*}_{L,s}:|\pi|=N\}\right|\geq\left|\{\pi\in D_{L,s}:|\pi|=N\}\right|$$

for every $N\geq M$. The source presents this as one of several related conjectures and notes that recent progress is discussed later in the paper; its resolution is not established by the supplied context.