Conjecture on the Laplacian eigenvalue distribution of graphs with given diameter

Let $G$ be a connected graph of order $n$ with diameter $d\ge 2$. For an interval of indices, write $m_G[a,b]$ for the number of Laplacian eigenvalues of $G$ whose indices lie between $a$ and $b$, inclusive.

Laplacian eigenvalue distribution conjecture. If $c=0,\dots,d-2$ and

$$\max\{2,c\}\le d\le n-2-c,$$

then

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

The conjecture concerns how many of the largest Laplacian eigenvalues a connected graph of diameter $d$ can have in the indicated index range. The source states that it is known for $c=0,1,2,d-3,d-2$, while the general case remains open.