The positive-real-part conjecture for $\mathfrak{L}_1$

Let $a>1$ and $\sigma>0$, and let $\mathfrak{L}_1$ be the linear operator for the first nonzero Fourier mode after reducing the eigenvalue problem from $\mathfrak{L}_n$ to $n=1$. Positive-real-part conjecture for $\mathfrak{L}_1$. For all $a>1$ and $\sigma>0$, all eigenvalues of $\mathfrak{L}_1$ have positive real parts. The conjecture is based on numerical evidence for the nonzero modes; the supplied text does not provide a proof or resolution.