Main parity conjecture for eta-quotients

Let

$$F(q)=\sum_{n\ge 0}c(n)q^n$$

be an eta-quotient, shifted by a suitable power of $q$ so that its powers are integral. Let $\delta_F$ denote the odd density of its coefficients. For a nonnegative integer-valued polynomial $P$ of positive degree, consider the sequence $c(P(m))$.

Main parity conjecture for eta-quotients. The following assertions hold: (i) for every $F$, $\delta_F$ exists and satisfies $\delta_F\le 1/2$; (ii) if $\delta_F=1/2$, then $c(P(m))$ has odd density $1/2$ for every such polynomial $P$, in particular on every arithmetic progression; (iii) if $\delta_F<1/2$, then the coefficients of $F$ vanish modulo $2$ identically on some arithmetic progression; and (iv) if the coefficients do not vanish modulo $2$ identically on any arithmetic progression, then they have odd density $1/2$ on every arithmetic progression, and hence $\delta_F=1/2$.

This is presented as a broad organizing conjecture for parity phenomena of eta-quotients, encompassing the alternatives between density $1/2$ and identically even arithmetic progressions. The source presents it as an open conjecture and notes that (i)--(iii) imply (iv), while (iv) implies (iii).