MakerBot Print is our newest print-prepration software, which supports native CAD files and STL assemblies,
allows you to interact with all your printers via the Cloud, and many other exciting new features.

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Solved myself(thanks, Google!)
Reason: S3D sees self intersecting surfaces (Repair - Identify self intersecting surfaces. It highlights the very edges I had problem with)
Solution: Mesh - Separate connected surfaces. It splits the model as several lillte models, but at least now it is sliced as expected