Infinite-matrix limit conjecture for the integral means spectrum of whole-plane LLEs

Let a whole-plane LLE be driven by a Lévy process without drift. For each $N$, let $B_N$ be the $N$-dimensional matrix obtained from the three-diagonal matrix $B$ in the unbounded version of LLE, or from the corresponding matrix in the bounded version, and let $\beta_{\rm \max}(N)$ be its maximal real eigenvalue. Equivalently, consider the maximal roots of the polynomials defined by the corresponding recurrence relations. Infinite-matrix limit conjecture. The value $\beta(2)$ is the limit of the sequence $\beta_{\rm \max}(N)$ as $N\to\infty$. The finite-truncation case is supported by numerical checks for SLE$_\kappa$ and PLE$_\lambda$, but convergence in the general whole-plane LLE setting remains open.