  Integer partitions have long been of interest to number theorists, perhaps
most notably Ramanujan, and are related to many areas of mathematics including
combinatorics, modular forms, representation theory, analysis, and mathematical
physics. Here, we focus on partitions with gap conditions and partitions with
parts coming from fixed residue classes.
  Let $\Delta_d^{(a,b)}(n) = q_d^{(a)}(n) - Q_d^{(b)}(n)$ where $q_d^{(a)}(n)$
counts the number of partitions of $n$ into parts with difference at least $d$
and size at least $a$, and $Q_d^{(b)}(n)$ counts the number of partitions into
parts $\equiv \pm b \pmod{d + 3}$. In 1956, Alder conjectured that
$\Delta_d^{(1,1)}(n) \geq 0$ for all positive $n$ and $d$. This conjecture was
proved partially by Andrews in 1971, by Yee in 2008, and was fully resolved by
Alfes, Jameson and Lemke Oliver in 2011. Alder's conjecture generalizes several
well-known partition identities, including Euler's theorem that the number of
partitions of $n$ into odd parts equals the number of partitions of $n$ into
distinct parts, as well as the first of the famous Rogers-Ramanujan identities.
  In 2020, Kang and Park constructed an extension of Alder's conjecture which
relates to the second Rogers-Ramanujan identity by considering
$\Delta_d^{(a,b,-)}(n) = q_d^{(a)}(n) - Q_d^{(b,-)}(n)$ where $Q_d^{(b,-)}(n)$
counts the number of partitions into parts $\equiv \pm b \pmod{d + 3}$
excluding the $d+3-b$ part. Kang and Park conjectured that
$\Delta_d^{(2,2,-)}(n)\geq 0$ for all $d\geq 1$ and $n\geq 0$, and proved this
for $d = 2^r - 2$ and $n$ even.
  We prove Kang and Park's conjecture for all but finitely many $d$. Toward
proving the remaining cases, we adapt work of Alfes, Jameson and Lemke Oliver
to generate asymptotics for the related functions. Finally, we present a more
generalized conjecture for higher $a=b$ and prove it for infinite classes of
$n$ and $d$.
