The dimension quotient conjecture for dihedral quandle rings

Let $\operatorname{R}_n$ be the dihedral quandle and let $R$ be a ring. For the quandle ring $R[\operatorname{R}_n]$, let $\Delta(\operatorname{R}_n)$ denote its augmentation ideal and let $\Delta^k(\operatorname{R}_n)$ denote its $k$-th power. Dimension quotient conjecture.

1. If $n>1$ is an odd integer, then

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

for all $k \geq 1$.

2. If $n>2$ is an even integer, then

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

for all $k \geq 2$.

The preceding computations for small dihedral quandles motivate these formulas for the successive quotients of the augmentation filtration. The statement predicts uniform behavior for all odd and even values of $n$, but no resolution is supplied in the given text.