The three-part classification conjecture for countable direct sums of chains

Let a DSC be a direct sum of chains, and let $Sib(\mathcal{D})$ denote its sibling number. Suppose

$$\mathcal{D}=\mathcal{D}_1\oplus\mathcal{D}_2\oplus\mathcal{D}_3$$

is a countable DSC, where $\mathcal{D}_1$ is bounded, $\mathcal{D}_2$ is a non-empty finite direct sum of chains with sibling number $\aleph_0$, and $\mathcal{D}_3$ is a direct sum of chains with finitely many non-trivial components, each having one sibling. Three-part classification conjecture. Then

$$Sib(\mathcal{D})=\aleph_0.$$

This is presented as a proposed form for the countable direct sums of chains with sibling number $\aleph_0$; the supplied context does not state that it has been proved.