The parity-dependent spectrum conjecture for repeated-part type-A seaweeds

For integers $k,r\geq 1$, let

$$\mathfrak{g}=\mathfrak{p}^A\frac{\overbrace{2k|\cdots|2k}^r|1}{2kr+1}.$$

Parity-dependent spectrum conjecture. The seaweed $\mathfrak{g}$ is Frobenius, and the set of distinct eigenvalues in its spectrum is the collection of integers in

$$\begin{cases} [-2k+1,2k], & \text{if } r \text{ is odd};\\ [-k,k+1], & \text{if } r \text{ is even}. \end{cases}$$

Moreover, $\mathfrak{g}$ has the unimodal spectrum property. This conjecture records a further stable family of spectra; the paper gives related computations and notes that the family need not have the log-concave spectrum property, while the stated general assertion remains open.