Inagaki–Tamura shift identity conjecture for partition inequalities

Let $p(n\mid\text{condition})$ count partitions of $n$ satisfying the specified condition. Define

$$q_d^{(1)}(n):=p(n\mid\text{parts}\geq 1\text{ and differ by at least }d)$$

and

$$Q_d^{(1,-)}(n):=p(n\mid\text{parts}\equiv\pm1\pmod{d+3},\text{ excluding the part }d+2).$$

Inagaki–Tamura shift identity conjecture. If $d\geq 12$ and $n\geq d+2$, then

$$q_d^{(1)}(n)-Q_{d-4}^{(1,-)}(n)\geq 0.$$

Shift identities of this type can yield information about the modified Alder-type inequalities and improve the bounds in the generalized conjecture. The source gives no resolution of this particular shift identity conjecture.