The even-multiplicity descent conjecture for decomposition lengths

Let $S$ be a numerical semigroup of multiplicity $m=2k$, where $k\in\mathbb{Z}$. A decomposition of $S$ is an irredundant intersection of irreducible numerical semigroups, and its length is the number of factors. Even-multiplicity descent conjecture. If $S$ has a decomposition of length $j$ with $k<j<m$, then $S$ has a decomposition of length $j-1$. This would give an additional restriction on the possible decomposition lengths for numerical semigroups of even multiplicity, and computational evidence in the surrounding discussion supports the related multiplicity-eight case. The conjecture is not proved in the supplied text.