Conway and Croft show it can be done for closed intervals and cannot be done for open intervals in the paper:
Covering a sphere with congruent great-circle arcs. Proc. Cambridge Philos. Soc. 60, 1964, pp787–800.
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The original answer somehow highlighted bibliography details of the article as a link but to nowhere. I decided to make the details correctly presented and add the actual link to the paper.
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