Alder-type partition inequality at the general level

A Known Alder-type partition inequality of level $a$, which involves the second Rogers-Ramanujan identity when the level $a$ is 2, states that the number of partitions of $n$ into parts differing by at least $d$ with the smallest part being at least $a$ is greater than or equal to that of partitions of $n$ into parts congruent to $\pm a \pmod{d+3}$, excluding the part $d+3-a$. In this paper, we prove that for all values of $d$ with a finite number of exceptions, an arbitrary level $a$ Alder-type partition inequality holds without requiring the exclusion of the part $d+3-a$ in the latter partition.