Local Tor-length conjecture for regular sequences

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 $k\leq n-1$, and let $J$ be an ideal generated by a regular sequence of length $n$. Local Tor-length conjecture. For every $0\leq i\leq k$,

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

The paper introduces this as a more restrictive conjecture after presenting counterexamples to the preceding Tor-length conjecture, noting that it remains relevant to the complete-intersection-singularity application. Unlike the preceding formulation, no $\mathfrak{m}$-primary assumption on $I+J$ is included in the stated claim.