Bardakov–Passi–Singh conjecture on augmentation quotients of dihedral quandles

Let $\mathbb{R}_n=\{a_0,a_1,\cdots,a_{n-1}\}$ denote the dihedral quandle of order $n$, and let $\Delta^k(\mathbb{R}_n)$ denote the $k$th power of its augmentation ideal. Bardakov–Passi–Singh conjecture. The following statements should hold:

1. For an odd integer $n>1$,

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

for all $k\ge 1$. 2. For an even integer $n>2$,

$$\left|\Delta^{k}(\mathbb{R}_{n})/\Delta^{k+1}(\mathbb{R}_{n})\right|=n$$

for $k\ge 2$.

The first assertion has been confirmed, and the second is known for $n=4$. The paper gives a counterexample showing that the conjecture is false in general, so the combined conjecture is refuted.