Generic Hausdorff dimension conjecture for Takagi-function abscissa level sets

Let $\tau$ be the Takagi function, and for $x\in[0,1]$ let $L(\tau(x))$ denote the level set of points having Takagi value $\tau(x)$. A set of abscissa points has full Lebesgue measure if its complement in $[0,1]$ has Lebesgue measure zero.

Abscissa generic level sets. A full Lebesgue measure set of abscissa points $x\in[0,1]$ have level sets $L(\tau(x))$ that are uncountable and have Hausdorff dimension $0$.

This conjecture refines the known almost-everywhere result that generic local level sets are uncountable and have Hausdorff dimension zero, by asserting the same properties for the full level sets $L(\tau(x))$.