Singular function that is Holder for all $\alpha<1$

Question

I am asking for an example of a singular continuous function that is Holder for all $\alpha<1$.

We know that such function cannot be Lipschitz, otherwise it is absolutely continuous.

We also know that a standard Cantor function is singular, but it is not Holder for $\alpha>\log 2/\log 3$.

So I am guessing if there is some kind of "fat Cantor function" that works.