Improved non-bipartite median-eigenvalue bound

Prove or disprove that, for every integer d >= 4, the median eigenvalues of every connected graph of maximum degree d have absolute value at most sqrt(d - 1).

Background

Earlier work established that the median eigenvalues of every connected graph with maximum degree d have absolute value at most sqrt(d), and the bound was improved to sqrt(d - 1) when d = 3.

The authors state that the same improvement is conjectured, or considered believable, for every d >= 4. Establishing this would extend the d = 3 non-bipartite result to all higher maximum degrees.