It is known that you can “reparameterise” Osgood curves to make them almost-everywhere smooth curves (simply compose one after the Cantor function). However doing this breaks injectivity, stopping them from being Osgood curves anymore.

Is it possible to use some other kind of trick or construction, to build an Osgood curve which is almost-everywhere differentiable (and remains injective)?

I know the answer is false if you replace almost-everywhere differentiable with differentiable (or if you only allow countably many non-differentiable points).

Bonus: If the answer turns out to be no, is there any comment about how differntiable an Oscood curve can be? Specifically ones where every sub-arc is also an Oscood curve?

It is known that you can “reparametrize” Osgood curves to make them almost-everywhere smooth curves (simply compose one after the Cantor function). However doing this breaks injectivity, stopping them from being Osgood curves anymore.

Is it possible to use some other kind of trick or construction, to build an Osgood curve which is almost-everywhere differentiable (and remains injective)?

I know the answer is false if you replace almost-everywhere differentiable with differentiable (or if you only allow countably many non-differentiable points).

Bonus: If the answer turns out to be no, is there any comment about how differentiable an Osgood curve can be? Specifically ones where every sub-arc is also an Osgood curve?