Ahanjideh–Akbari–Fakharan–Trevisan conjecture on Laplacian eigenvalues and diameter

Let $G$ be a connected graph of order $n$, let $d$ be its diameter with $d\ge 2$, and let $m_GI$ denote the number of Laplacian eigenvalues of $G$ in an interval $I\subseteq[0,n]$. The path on $d+1$ vertices is denoted by $P_{d+1}$. Ahanjideh–Akbari–Fakharan–Trevisan conjecture. If $G\ne P_{d+1}$, then

$$m_G[n-d+2,n]\le n-d.$$

This conjecture concerns the distribution of Laplacian eigenvalues of connected graphs in relation to their diameter. The stated source notes that it was verified when $d\in\{2,3\}$ or when the diameter is at most the independence number; its general status is not specified here.