A mixed graph \(G\) is said to be strongly determined by its generalized Hermitian spectrum (abbreviated SHDGS), if, up to isomorphism, \(G\) is the unique mixed graph that is cospectral with \(G\) w.r.t. the generalized Hermitian spectrum. The authors conjecture that every such graph is SHDGS and prove that, for any mixed graph \(H\) that is cospectral with \(G\) w.r.t. the generalized Hermitian spectrum, there exists a Gaussian rational unitary matrix \(U\) with \(Ue = e\) such that \(U^\ast A(G)U = A(H)\) and \((1+i)U\) is a Gaussian integral matrix. The authors verify the conjecture in two extremal cases when \(G\) is either an undirected graph or a self-converse oriented graph. Consequently, the authors prove that all directed paths of even order are SHDGS.