Nonnegative $q$-series product conjectures

Suppose $|q|<1$ and $z\ne0$. Here $E(q):=\prod_{n=1}^{\infty}(1-q^n)$ and $(x;q)_\infty:=\prod_{n=0}^{\infty}(1-xq^n)$. Write $F(q,z)\succeq0$ to mean that the relevant Laurent series in $q$ and $z$ has nonnegative coefficients.

Nonnegative product conjecture.

(i) If $p\ge1$, then

$$\frac{E(q)}{(z;q)_\infty(qz^{-p};q)_\infty}\succeq0.$$

(ii) If $a,b,m,n\ge1$, then

$$\frac{E(q^{ma+nb})}{(q^a;q^{ma+nb})_\infty(q^b;q^{ma+nb})_\infty}\succeq0.$$

(iii) For $a\ge1$,

$$\frac{[z^a;q]_\infty E(q)}{[z;q]_\infty[z^{a+1};q]_\infty}\succeq0.$$

These are further positivity conjectures for $q$-products, following preceding propositions on analogous inequalities. The supplied text gives no evidence that any of the three assertions has been resolved.