Higher-order modified Alder conjecture for a=3

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 $\pm b$ modulo $d+3$, excluding the part $d+3-b$, and define

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

Higher-order modified Alder conjecture. For all $d,n\geq 1$,

$$\Delta_d^{(3,-)}(n)\geq 0.$$

This is proposed as a generalization of Kang and Park's modified Alder conjecture to higher equal values of $a=b$. The source gives no proof or counterexample, so the assertion remains open.