The heat flow conjecture, as studied in the finite-dimensional random matrix setting, asks whether applying the heat semigroup to the characteristic polynomial of a large random matrix preserves or transforms the zero distribution in a predictable and tractable way. The natural next question is whether an analogous statement holds when the ambient space is infinite-dimensional, such as when working with random operators on a Hilbert or nuclear space, where characteristic polynomials no longer exist in a classical sense but spectral zeta functions or Fredholm determinants serve as replacements. Specifically, one wants to know if the zeros of these spectral functions under heat flow converge to a well-defined limiting distribution that mirrors the finite-dimensional picture.