Waring conjecture for middle-$\frac{1}{\alpha}$ Cantor sets

Let $\alpha>1$, set

$$r=\frac12\left(1-\frac1\alpha\right),$$

and let $C_\alpha$ be the middle-$\frac1\alpha$ Cantor set. For any real number $m\geq1$, define the integer

$$k=\left\lceil\left(\frac1r-1\right)^m\right\rceil.$$

Waring conjecture for $C_\alpha$.

$$[0,k]=\left\{x_1^m+x_2^m+\cdots+x_k^m:x_j\in C_\alpha,\ 1\leq j\leq k\right\}.$$

This conjecture extends the paper's Waring-type representation problem from the Cantor ternary and middle-half cases to general middle-$\frac1\alpha$ Cantor sets. The source does not state a resolution.