Peirce-spectrum conjecture for complex algebras of odd dihedral quandles

Let $R_n$ be a dihedral quandle of odd order $n$, let $\mathbb{C}[R_n]$ be its complex quandle algebra, and let $\sigma(\mathbb{C}[R_n])$ denote its Peirce spectrum. Peirce-spectrum conjecture. The Peirce spectrum of the complex quandle algebra of a dihedral quandle of odd order is

$$\sigma(\mathbb{C}[R_n])=\{0,1,-1\}.$$

For the examples discussed in the paper, namely $R_3$, $R_5$, $R_7$, and $R_9$, the computed Peirce spectra have this form; the assertion for every odd-order dihedral quandle remains open.