The even-regular partition density conjecture

Let $b_m(n)$ be the coefficients of the generating function for $m$-regular partitions. For even $m$, write

$$m=2^j m_0,$$

where $m_0$ is odd. The odd density of a sequence is the limiting proportion of its terms that are odd, when the limit exists; a sequence is lacunary mod $2$ when its odd density is zero. Even-regular partition density conjecture. If $2^j>m_0$, then $b_m$ is identically even on infinitely many nonconstant, non-nested subprogressions $An+B$, in addition to being lacunary mod $2$. If $2^j<m_0$, then the odd density of $b_m$ is $1/2$, and the odd density on any nonconstant subprogression $An+B$ is also $1/2$; in particular, no nonconstant subprogression is identically even. The first regime incorporates the cited lacunarity theorem, while the second predicts maximal odd density both globally and on every nonconstant arithmetic progression. The source illustrates the first regime with a proved example for $b_{40}$, but the full conjecture remains unresolved in the supplied text.