Proof of Vivo-Pato-Oshanin's conjecture on the fluctuation of von Neumann entropy

It was recently conjectured by Vivo, Pato, and Oshanin [Phys. Rev. E 93, 052106 (2016)] that for a quantum system of Hilbert dimension mn in a pure state, the variance of the von Neumann entropy of a subsystem of dimension m <= n is given by -psi(1)(mn+1) + m+n/mn+1 psi(1)(n) - (m+1)(m+2n+1)/4n(2()mn+1), where psi(1)(.) is the trigamma function. We give a proof of this formula.