Conjectured quartic-invariant inequalities for the distinct partition function

Let $q(n)$ denote the number of partitions of $n$ into distinct parts, and set $a_n=q(n)$. For a quartic binary form, define

$$A(a_0,a_1,a_2,a_3,a_4)=a_0a_4-4a_1a_3+3a_2^2,$$ $$B(a_0,a_1,a_2,a_3,a_4)=-a_0a_2a_4+a_2^3+a_0a_3^2+a_1^2a_4-2a_1a_2a_3,$$ $$I(a_0,a_1,a_2,a_3,a_4)=A(a_0,a_1,a_2,a_3,a_4)^3-27B(a_0,a_1,a_2,a_3,a_4)^2.$$

The quartic-invariant conjecture for $q(n)$. The following inequalities hold:

$$A(a_{n-1},a_n,a_{n+1},a_{n+2},a_{n+3})>0\quad\text{for }n\geq230,$$ $$B(a_{n-1},a_n,a_{n+1},a_{n+2},a_{n+3})>0\quad\text{for }n\geq272,$$

and

$$I(a_{n-1},a_n,a_{n+1},a_{n+2},a_{n+3})>0\quad\text{for }n\geq267.$$

These are proposed as analogues for the distinct partition function of quartic-binary-form invariant inequalities previously studied for the ordinary partition function. The supplied text gives no resolution of these conjectures.