Noncongruence and odd-density conjecture for eta-quotient coefficients

Let $f_m$ and $b_m(n)$ denote the eta-quotient and its coefficient sequence used in the paper, respectively, and let the odd density of a series mean the limiting proportion of its coefficients that are odd. Fix

$$m \in \{6,10,14,15,18,20,22,26,27,28\}.$$

Noncongruence and odd-density conjecture. The following assertions hold:

1. There are no integers $A>0$ and $B\geq 0$ such that

$$b_m(An+B)\equiv 0 \pmod{2}$$

for all $n$. 2. The series $f_m/f_1$ has odd density $1/2$.

These claims predict that the indicated coefficient sequences have no identically even arithmetic progression, while the associated eta-quotient has asymptotically half of its coefficients odd. They are presented as noncongruences analogous to results of Ono and Radu for the partition function; their status is unresolved in the supplied source.