The Lech-type Dutta-multiplicity conjecture for complete local rings

Let $R$ be a complete local ring, and let $F$ be a short complex in the perfect-complex category, meaning a short complex in $\operatorname{perf} R$ supported on the maximal ideal. Let $e(R)$ denote the Hilbert–Samuel multiplicity of $R$, and let $\chi_\infty(F)$ denote the Dutta multiplicity. Lech-type Dutta-multiplicity conjecture. If $F$ is a short complex in $\operatorname{perf} R$, then

$$\chi_\infty(F)\geq e(R).$$

This is proposed as an extension of the preceding positive-characteristic results and is presented in the context of a reformulation of Lech's conjecture. Its status is not resolved in the supplied text.