Bressoud's partition identity conjecture for even moduli

Let $A_j(\alpha_1,\ldots,\alpha_\lambda;\eta,k,r;n)$ count partitions of $n$ into parts congruent to $0,\alpha_1,\ldots,\alpha_\lambda\pmod{\eta}$ satisfying Bressoud's stated congruence and multiplicity conditions, and let $B_j(\alpha_1,\ldots,\alpha_\lambda;\eta,k,r;n)$ count partitions of $n$ satisfying the corresponding difference, multiplicity, initial-condition, and parity conditions. For $j=0$ or $1$, $(2k+j)/2>r\geq\lambda\geq0$, and $n\geq0$, these two partition functions are equal:

Bressoud's conjecture.

$$A_j(\alpha_1,\ldots,\alpha_\lambda;\eta,k,r;n)=B_j(\alpha_1,\ldots,\alpha_\lambda;\eta,k,r;n).$$

This conjecture proposes an overpartition-free partition identity equating a congruence-side enumeration with a difference-condition enumeration for the indicated even-modulus parameters. Its resolution is not established by the supplied source context.