In 2020, Kang and Park conjectured a "level 2" Alder-type partition
inequality which encompasses the second Rogers-Ramanujan Identity. Duncan,
Khunger, the fourth author, and Tamura proved Kang and Park's conjecture for
all but finitely many cases utilizing a "shift" inequality and conjectured a
further, weaker generalization that would extend both Alder's (now proven) as
well as Kang and Park's conjecture to general level. Utilizing a modified shift
inequality, Inagaki and Tamura have recently proven that the Kang and Park
conjecture holds for level 3 in all but finitely many cases. They further
conjectured a stronger shift inequality which would imply a general level
result for all but finitely many cases. Here, we prove their conjecture for
large enough n, generalize the result for an arbitrary shift, and discuss the
implications for Alder-type partition inequalities