Odd density conjecture for residue classes of 21-regular partitions

Let $b_{21}(n)$ be the number of $21$-regular partitions of $n$.

Odd density conjecture for $b_{21}$. The series

$$\sum_{n=0}^\infty b_{21}(4n+2)q^n$$

and

$$\sum_{n=0}^\infty b_{21}(4n+3)q^n$$

have odd density $1/2$. Consequently, the $21$-regular partition function has odd density $1/4$.

The preceding arguments establish lacunarity modulo $2$ for the residue classes $4n$ and $4n+1$, while this conjecture predicts the contrasting density-$1/2$ behavior of the other two classes. The source gives no resolution.