Scaling of ground-state overlap for classically preparable trial states
In plain words
Quantum phase estimation needs a starting state that resembles the true ground state. If the best starting states that ordinary computers can prepare become exponentially worse as molecules grow, the quantum speedup disappears.
Precise statement
For strongly correlated molecules and clusters (e.g. iron-sulfur clusters with $N_{\mathrm{Fe}}$ iron atoms), determine how the maximal overlap $\mid\langle \phi \mid \psi_0\rangle\mid^2$ between the ground state $\psi_0$ and states $\phi$ preparable classically in poly time (bounded-bond-dimension MPS, broken-symmetry determinants and short sums of them) scales with system size; exponential decay would remove the exponential advantage of phase estimation.
What would settle it
Computed overlaps for a size series of a chemically relevant family large enough to fit the scaling, or a proof for a model family.