I have proposed very recently a question in the following link concerning the title of the current question: Difference of the maximum eigenvalue of a graph with the one of one-edge-deleted subgraph

I would like to ask my new related question as follows:

Is there a positive function $f(n,m)$ such that for any graph $G$ with $n$ vertices and $m$ edges and any edge $e$ such that $G\setminus e$ is connected the inequality $\lambda(G)-\lambda(G\setminus e)\geq f(n,m)$ holds?

The notations are as in the above link. According to Prof. Chris Godsil's answer to my question posed in the above link one must have $f(n,m)\leq \frac{\pi^2}{(n+1)^2}$.

Edit: thanks to Anthony, I add the assumption $G\setminus e$ is connected.