I’ve proved this conjecture (not a long standing conjecture, it took just one day to solve it) and found a stronger theorem than these propositions.

So my new theorem:

Theorem $latex (\mathsf{FCD})$ and $latex (\mathsf{RLD})_{\mathrm{out}}$ form mutually inverse bijections between complete reloids and complete funcoids.

For a proof see this note.

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