The conjectured upper bound for the largest Neumann eigenvalue of a path graph

Let $P$ be a path graph with $n$ vertices, and let $\lambda_{n-2}(P)$ denote its largest Neumann eigenvalue. Path-graph Neumann eigenvalue conjecture. The largest Neumann eigenvalue satisfies

$$\lambda_{n-2}(P)\leq 4.$$

This conjecture proposes a sharp uniform upper bound for the largest Neumann eigenvalue of path graphs. The supplied text gives no resolution, so its status remains open.