BPS conjecture on augmentation quotients of dihedral quandle rings

Let $R_n$ be the dihedral quandle, and let $\Delta^k(R_n)$ denote the $k$-th power of its augmentation ideal.

BPS conjecture. \begin{enumerate} \item If $n>1$ is an odd integer, then

$$\Delta^k(R_n) / \Delta^{k+1}(R_n) \cong \mathbb{Z}_n$$

for all $k \geq 1$. \item If $n>2$ is an even integer, then

$$|\Delta^k(R_n) / \Delta^{k+1}(R_n)| = n$$

for all $k \geq 2$. \end{enumerate} The conjecture concerns the successive quotients of powers of the augmentation ideal in the quandle ring of a dihedral quandle. The source paper presents it as a conjecture from BPS and discusses its proof and further progress, but the supplied status is unknown.