I don't see any open paths in the area where these straight lines are happening. The only somewhat odd thing about the file, is that is contains 2 objects. One is the design that you've drawn. And the other is a Group of 0 (zero). I suppose that could either be confusing Inkscape when you save as EPS, or confusing the cutter?
By using the Tab key, you can select each object in a file, consecutively. So Tab (selects your drawing) and Tab again, selects the Group of 0. Be sure to watch the status bar to make sure when the Group of 0 is selected. Then press Delete.
Then try saving as EPS, and loading the new EPS into the cutter.
Another thing you can try, is when you save as EPS, Inkscape shows a little dialog with a few options. One of those is a choice between Post Script level 2 and Post Script level 3. So whichever one you used the first time, try the other one, and see if that helps. Your machine may be expecting one or the other.
There are some other options in that dialog. They don't look like they would cause this problem to me, but if this problem is still persisting, you can use your judgement, and change them if you think best.
If none of that helps, I guess I would look towards the software for your cutter.