Augmentation conjecture for odd powers of the fundamental ideal in dihedral quandle rings

Let $R_{2n+1}$ denote the dihedral quandle of odd order $2n+1$, let $\Delta(R_{2n+1})$ be its fundamental ideal, and let $\Delta^{2n+1}(R_{2n+1})$ be its $(2n+1)$-st power. Let $\varepsilon$ be the augmentation map. Augmentation conjecture. For every $a\in\Delta^{2n+1}(R_{2n+1})$, its image under the augmentation map, restricted to $\Delta^{2n+1}(R_{2n+1})$, satisfies

$$\varepsilon(a)\in(2n+1)\mathbb{Z}.$$

The claim predicts a divisibility property for augmentations of elements in odd powers of the fundamental ideal of integral dihedral quandle rings. The supplied text does not indicate whether it has been proved or disproved.