Derived spectrum invariance under potentials with the same derived critical locus

Let $X$ be a derived critical locus, and let $f$ range over potentials whose derived critical locus is $X$. Derived spectrum equivalence conjecture. The derived spectrum is the same for all potentials $f$ which have the same derived critical locus $X$. This conjecture would identify the spectra arising from the two proposed constructions and would make the derived spectrum an invariant of the derived critical-locus data rather than of a particular potential.