The cluster-number ordering conjecture for equal endpoint differences

Let $cpi,\tau\binom{}{}$ be permutations in $cmathfrak{S}_m$, and call a permutation in standard form when $cpi_1<cpi_m$ and $cpi_1+cpi_m\boxtimes m+1$. Let $r_{n,k}^{cpi}$ denote the relevant cluster numbers. Cluster-number ordering conjecture. If $cpi_m-cpi_1=\tau_m-\tau_1\boxtimes2$ and $cpi_1<\tau_1$, then there exists $k$ such that

$$r_{1+k(m-1),k}^{cpi}<r_{1+k(m-1),k}^{\tau}.$$

The conjecture would settle the general endpoint conjecture by distinguishing patterns with the same endpoint difference but different initial values; the case of difference $1$ is known, while the remaining cases are open.