The complete-intersection semigroup-condition conjecture for splice diagrams

Let $X$ be a surface singularity with homology sphere link, and let $\Delta$ be its splice diagram. For each node $v$ and adjacent edge $e$, the semigroup condition requires the edge weight $d_{ve}$ to lie in the semigroup generated by the relevant linking numbers:

$$d_{ve}\in\mathbb{N}\langle\ell'_{vw}:w\text{ a leaf of }\Delta\text{ in }\Delta_{ve}\rangle.$$

Complete-intersection semigroup-condition conjecture. If $X$ is a complete intersection, then $\Delta$ satisfies the semigroup condition.

If true, this would make the combinatorial semigroup condition necessary as well as sufficient for constructing complete intersection singularities of splice type with prescribed homology-sphere links. The paper reports no counterexample.