The generalized Freyd conjecture for a triangulated category

Let $\mathscr{T}$ be a triangulated category, let $\mathscr{B}$ be a small full subcategory closed under the shift functor $\Sigma$, and let $\mathscr{C}$ be the thick full subcategory generated by $\mathscr{B}$. Write $[X,Y]$ for the abelian group of maps from $X$ to $Y$ in $\mathscr{T}$, and let $\mathscr{P}\mathscr{B}$ and $\mathscr{P}\mathscr{C}$ be the categories of additive functors from $\mathscr{B}^{op}$ and $\mathscr{C}^{op}$ to $\mathsf{Ab}$. Define the Freyd functor $\mathbb{F}\colon\mathscr{T}\to\mathscr{P}\mathscr{B}$ by $\mathbb{F}X(-)=[-,X]$ and $\mathbb{F}f=[-,f]$. The generalized Freyd conjecture. The restriction $\mathbb{F}\colon\mathscr{C}\to\mathscr{P}\mathscr{B}$ is faithful; equivalently, for every map $f$ in $\mathscr{C}$, $\mathbb{F}f=0$ if and only if $f=0$. This is a generalization of the generating hypothesis for the stable homotopy category, but it is false without additional hypotheses; the paper studies formal conditions under which it holds.