Mohar's HL-index conjecture for planar subcubic graphs

Let $G$ be a simple graph of order $n$, and let its adjacency eigenvalues be

$$\lambda_1(G)\ge\dots\ge\lambda_n(G).$$

A graph is subcubic if its maximum degree is at most $3$, and it is planar if it can be drawn in the plane without edge crossings. Define the HL-index by

$$R(G)=\max\left\{|\lambda_h(G)|,|\lambda_l(G)|\right\},$$

where

$$h=\left\lfloor\frac{n+1}{2}\right\rfloor,\qquad l=\left\lceil\frac{n+1}{2}\right\rceil.$$

Mohar's HL-index conjecture. For every subcubic planar graph $G$,

$$R(G)\le 1.$$

Mohar proved the weaker bound $R(G)\le\sqrt{2}$ for every subcubic graph. The conjecture asserts the sharper bound for planar subcubic graphs and remains unresolved based on the supplied text.