Piecewise linearity of the Freiman dimension function and constancy of the covering function

Let $F(K)$ and $G(K)$ be the functions introduced earlier in the paper for measuring the linear structure of finite binary sets with doubling parameter $K$. The piecewise-linearity and piecewise-constancy conjecture. For $K\geq 1$,

$$F(K)\text{ is a piecewise linear function, and }G(K)\text{ is a piecewise constant function.}$$

This conjecture proposes a global structural description of the two functions arising in the study of small-doubling binary sets. The supplied context does not define $F(K)$ and $G(K)$ or provide evidence concerning the conjecture's resolution.