Four-square representation conjecture for the middle-thirds Cantor set

Let $C$ denote the middle-thirds Cantor set. For a real number $u\in[0,1]$, consider representations of $u$ as a sum of four squares of elements of $C$. Four-square representation conjecture. Every $u\in[0,1]$ can be written as

$$u=x_1^2+x_2^2+x_3^2+x_4^2,$$

where $x_i\in C$ for $1\leq i\leq 4$. The conjecture is supported by strong numerical evidence, but no proof or disproof is given here.