The recursive spectrum conjecture for repeated-part type-A seaweeds

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

$$\mathfrak{g}_{k,r}=\mathfrak{p}^A\frac{\overbrace{2k|\cdots|2k}^r|1}{1|\underbrace{2k|\cdots|2k}_r}.$$

Recursive spectrum conjecture. The distinct eigenvalues in the spectrum of $\mathfrak{g}_{k,r}$ are precisely the integers in $[-k+1,k]$, and $\mathfrak{g}_{k,r}$ has the log-concave spectrum property. Moreover, if $i\in[-k+1,0]$, then the multiplicity of $i$ in the spectrum of $\mathfrak{g}_{k,r}$ equals the multiplicity of $i-1$ in the spectrum of $\mathfrak{g}_{k+1,r}$; similarly, if $i\in(0,k]$, then the multiplicity of $i$ in the spectrum of $\mathfrak{g}_{k,r}$ equals the multiplicity of $i+1$ in the spectrum of $\mathfrak{g}_{k+1,r}$. This conjecture predicts both log-concavity and a precise recursion for multiplicities across the parameter $k$; it is presented on the basis of results and experimental evidence in the paper and remains open.