Lima–Pereira–Nuño–Ballesteros–Orefice–Okamoto–Tomazella Tor-length conjecture

Let $(R,\mathfrak{m})$ be a regular local ring of dimension $n$. Let $I$ be an ideal generated by an $R$-regular sequence of length $k\leq n$, and let $J$ be an ideal generated by a regular sequence of length $n$. Assume that $I+J$ is $\mathfrak{m}$-primary, and let $\ell$ denote the length function on $R$-modules. Lima–Pereira–Nuño–Ballesteros–Orefice–Okamoto–Tomazella conjecture. For every $0\leq i\leq k$,

$$\ell\left(\operatorname{Tor}_i^R(R/I,R/J)\right)=\binom{k}{i}\ell\left(\frac{R}{I+J}\right).$$

This conjecture is a local Tor-length variant of the Buchsbaum–Eisenbud–Horrocks conjecture arising in the study of complete-intersection singularities. The paper states that the corresponding conjecture fails in general, while the displayed formulation is the original version being discussed.