The bold conjecture on linear-data entailment

Let $\Phi$ and $\Psi$ be linear data, and let $\gamma\colon I^\Psi\to I^\Phi\times\{0,1\}$ be a partial function. Assume that whenever functions $F_i\colon W_i^\Phi\to\mathbb{R}/\mathbb{Z}$ and $A\in\mathbb{R}/\mathbb{Z}$ satisfy

$$\forall v\in V^\Phi\colon \sum_{i\in I^\Phi}F_i\bigl(\phi_i^\Phi(v)\bigr)=A,$$

there are functions $F'_j$ and $B\in\mathbb{R}/\mathbb{Z}$ satisfying

$$\forall u\in V^\Psi\colon \sum_{j\in I^\Psi}F'_j\bigl(\phi_j^\Psi(u)\bigr)=B,$$

with $F'_j=(-1)^\ell F_i$ whenever $\gamma(j)=(i,\ell)$ is defined. Bold conjecture. Under these assumptions, there is some $M\geq0$ such that $\Phi\vdash_\gamma^M\Psi$. This proposes that every valid functional-equation implication encoded by the linear data can be derived through the paper's Cauchy–Schwarz entailment system; the source presents it as the most ambitious version and gives no resolution.