Kang and Park's modified Alder conjecture

Let $q_d^{(a)}(n)$ count partitions of $n$ into parts at least $a$ whose parts differ by at least $d$, and let $Q_d^{(2,-)}(n)$ count partitions of $n$ into parts congruent to $7 2$ modulo $d+3$, excluding the part $d+1$. Define

$$\Delta_d^{(2,-)}(n):=q_d^{(2)}(n)-Q_d^{(2,-)}(n).$$

Kang and Park's conjecture. For all $d,n\geq 1$,

$$\Delta_d^{(2,-)}(n)\geq 0.$$

Kang and Park proved the conjecture when $d=2$ or $d=2^s-2$ for any positive integer $s\geq 5$, and for positive even $n$. The source paper proves it for $d\geq 62$ and all $n\geq 1$; the cases $d=1$ and $3\leq d\leq 61$ remain unresolved in the source.