The generalized double-exclusion Alder conjecture

Let $q_d^{(a)}(n)$ count partitions of $n$ into parts at least $a$ whose parts differ by at least $d$. For $1\leq b\leq d+2$, let $Q_d^{(b,-,-)}(n)$ count partitions of $n$ into parts congruent to $\pm b \pmod{d+3}$, excluding the parts $b$ and $d+3-b$. Define

$$\Delta_d^{(a,b,-,-)}(n)=q_d^{(a)}(n)-Q_d^{(b,-,-)}(n),\qquad \Delta_d^{(a,-,-)}(n)=\Delta_d^{(a,a,-,-)}(n).$$

Generalized double-exclusion Alder conjecture. Let $a,d$ be positive integers with $1\leq a\leq d+2$. Then for all $n\geq 1$,

$$\Delta_d^{(a,-,-)}(n)\geq 0.$$

This conjecture is proposed to extend the modified Alder inequality to arbitrary admissible $a$, after observing that the single-exclusion version can fail for $a\geq 4$. Its resolution is not given in the source.