The eta-quotient parity master conjecture

Let

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

be an eta-quotient shifted so that all powers of $q$ are integral, and let $\delta_F$ be the odd density of its coefficients $c(n)$, when this density exists. A polynomial is nonnegative integer-valued if it takes nonnegative integer values at the relevant integers. Eta-quotient parity master conjecture. For every $F$, $\delta_F$ exists and satisfies $\delta_F\leq 1/2$. If $\delta_F=1/2$, then for every nonnegative integer-valued polynomial $P$ of positive degree, the odd density of $c(P(n))$ is $1/2$; in particular, every nonconstant subprogression $c(An+B)$ has odd density $1/2$. If $\delta_F<1/2$, then the coefficients of $F$ are identically even on some nonconstant subprogression. If the coefficients are not identically even on any nonconstant subprogression, then they have odd density $1/2$ on every nonconstant subprogression, and hence $\delta_F=1/2$. The source notes the logical implications among these clauses and presents the conjecture as a master framework for parity questions about eta-quotients; it remains open in the supplied text.