Diameter vs Laplacian eigenvalue distribution

Let $G$ be a simple graph of order $n$. It is known that any Laplacian eigenvalue of $G$ belongs to the interval $[0,n]$. For an interval $I\subseteq [0, n]$, denote by $m_GI$ the number of Laplacian eigenvalues of $G$ in $I$, counted with multiplicity. When $G$ is connected, known results on the Laplacian eigenvalue distribution related to the diameter $d$ of $G$ include: $m_G[n-d+2,n]\le n-d$ if $2\le d\le n-3$ and $m_G[n-d+1,n]\le n-d+1$ if $1\le d\le n-3$. In this paper, we show that $m_G[n-d,n]\le n-d+2$ if $2\le d\le n-4$, and $m_G[n-2d+4,n]\le n-2$ if $2\le d\le \lfloor\frac{n}{2} \rfloor$.