Hagendorf Orders

Call a linear order type $φ$ Hagendorf if it shares two properties with additively indecomposable ordinal numbers without being one itself: $φ$ is strictly indecomposable to the right and whenever $ψ< φ$, then $ψ$ can be embedded into a proper initial segment of $φ$. In the 1970s, J. Hagendorf ask