The encoding conjecture for convex-formulation languages

Let $\mathbf{CF}$ be the class of languages $L\subseteq\{0,1\}^*$ whose length-$n$ slices have convex hulls admitting extended formulations of polynomial size, and let $\mathbf{CF}^{\operatorname{enc}}$ be the subclass admitting such formulations with integral coefficients of polynomial encoding length.

Encoding conjecture.

$$\mathbf{CF}^{\operatorname{enc}}=\mathbf{CF}.$$

The inclusion $\mathbf{CF}^{\operatorname{enc}}\subseteq\mathbf{CF}$ is immediate, while the reverse inclusion would say that every polynomial-size extended formulation can be replaced by one with polynomially bounded encoding length. The source presents this as an open conjecture; its significance is that it would identify the unrestricted and efficiently encodable classes of languages described by polynomial-size extended formulations.