Graph polynomials are graph invariants that map graphs to polynomials. As an analogue of the famous Tutte polynomial for directed graphs, Chung and Graham (J. Comb. Theory Series B 65(2):273–290, 1995) define the cover polynomial \(C_G(x,y)\) . Bläser and Dell (Automata, Languages and Programming, pp. 801–812. Springer, Berlin, 2007) prove that evaluating the cover polynomial is \(\#\mathsf {P}\) -hard, except for the points \((0,0),(0,-1),(1,-1)\) . There the evaluation is easy. However, the graphs used in this reduction are not simple. Motivated by a connection between the cover polynomial and the drop polynomial for simple graphs, Chung and Graham (J. Comb. Theory Series B 126:62–82, 2017) conjecture that the cover polynomial is also hard when restricted to simple graphs (which do not allow parallel edges or loops). As our main result, we confirm this conjecture.