Higher-dimensional converse for mixed Buchsbaum–Rim multiplicities
Determine whether, for a quasi-unmixed Noetherian local ring of dimension d≥2 with infinite residue field and finite-colength submodules E₁,E₂ of a free module F, equality of all mixed Buchsbaum–Rim multiplicities, e_BR(E₁^{[q−i]},E₂^{[i]})=c for every 0≤i≤q where q=d+rank(F)−1, implies equality of integral closures \overline{E₁}=\overline{E₂} in F.
References
At present, the example over $k[[x,y]]/(xy)$ only disproves the implication \textup{(i)}$\Rightarrow$\textup{(ii)} when $d=1$. The example above does not settle Question~\ref{converse}, which asks whether the implication holds or fails in dimensions at least two.
\begin{question} Assume in addition that $R$ is quasi-unmixed, $d\geq2$, and the residue field is infinite. Let $E_1,E_2\subseteq F$ have finite colength. If
e_{\rm BR}(E_1{[q-i]},E_2{[i]})=c \qquad\text{for every }0\leq i\leq q, must one have
\overline{E_1}=\overline{E_2}?
\end{question}