Let $A,B$ denote square matrices such that $\lvert A_{ij}\rvert\le B_{ij}$ for all $i,j$, and denote the spectral radius by $\rho$. From the Gelfand spectral radius formula it is easy to see that $$\rho(A)\le\rho(\lvert A\rvert)\le\rho(B),$$ where $\lvert A\rvert$ is meant entrywise.

**Question:** What is known about the case of equality? Is it true that equality can only occur if $A_{ij}=e^{i\phi}B_{ij}$ for some $\phi\in[0,2\pi]$?