The heat flow conjecture predicts that applying the heat operator to the characteristic polynomial of a large random Hermitian matrix produces a polynomial whose zeros, after appropriate rescaling, converge to the eigenvalues of the original matrix in a precise sense. This has been studied for classical Hermitian ensembles such as GUE and GOE, where symmetry forces eigenvalues to be real. The genuinely open question is whether an analogous conjecture can be formulated and proved for non-Hermitian random matrix ensembles such as the Ginibre ensemble, where eigenvalues are scattered in the complex plane, and whether the heat flow still organizes zeros along curves or regions with identifiable limiting geometry.