Maximal-eigenvalue conjecture for the integral means spectrum of whole-plane LLEs

Let a whole-plane LLE be driven by a Lévy process without drift, and suppose there is an integer $N$ such that

$$\eta_N=N+2$$

in the unbounded version of LLE, or

$$\eta_N=N-2$$

in the bounded version. Let $B$ denote the corresponding three-diagonal matrix given by (B) or (Bext). Maximal-eigenvalue conjecture. The value of the integral means $\beta$-spectrum at $q=2$ equals the maximal real eigenvalue of $B$. This conjecture extends the results known without the smallness condition on $\delta\eta_n$ in the cases covered by the cited theorems, and proposes that the maximal-eigenvalue description holds for the stated whole-plane LLEs generally.