In their celebrated 1991 paper on the inverse eigenvalue problem for nonnegative matrices, Boyle and Handelman conjectured that if A A is an ( n + 1 ) × ( n + 1 ) (n+1)\times (n+1) nonnegative matrix whose nonzero eigenvalues are: λ 0 ≥ | λ i | \lambda _0 \geq |\lambda _i| , i = 1 , … , r i=1,\ldots ,r , r ≤ n r \leq \ n , then for all x ≥ λ 0 x \geq \lambda _0 , ( ∗ ) ∏ i = 0 r ( x − λ i ) ≤ x r + 1 − λ 0 r + 1 . \begin{equation} \prod _{i=0}^{r} (x-\lambda _i) \leq x^{r+1}-\lambda _0^{r+1}.\tag *{$(\ast )$} \end{equation} To date the status of this conjecture is that Ambikkumar and Drury (1997) showed that the conjecture is true when 2 ( r + 1 ) ≥ ( n + 1 ) 2(r+1)\geq (n+1) , while Koltracht, Neumann, and Xiao (1993) showed that the conjecture is true when n ≤ 4 n\leq 4 and when the spectrum of A A is real. They also showed that the conjecture is asymptotically true with the dimension. Here we prove a slightly stronger inequality than in ( ∗ ) (\ast ) , from which it follows that the Boyle–Handelman conjecture is true. Actually, we do not start from the assumption that the λ i \lambda _i ’s are eigenvalues of a nonnegative matrix, but that λ 1 , … , λ r + 1 \lambda _1,\ldots , \lambda _{r+1} satisfy λ 0 ≥ | λ i | \lambda _0\geq |\lambda _i| , i = 1 , … , r i=1,\ldots , r , and the trace conditions: ( ∗ ∗ ) ∑ i = 0 r λ i k ≥ 0 , for all k ≥ 1. \begin{equation} \sum _{i=0}^{r} \lambda _i^k \geq 0, \ \mbox {for all} k \geq 1.\tag *{$(\ast \ast )$} \end{equation} A strong form of the Boyle–Handelman conjecture, conjectured in 2002 by the present authors, says that ( ∗ * ) continues to hold if the trace inequalities in ( ∗ ∗ ** ) hold only for k = 1 , … , r k=1,\ldots ,r . We further improve here on earlier results of the authors concerning this stronger form of the Boyle–Handelman conjecture.