Hom boundedness implies cone finiteness

A triangulated category $\mathsf{D}$ is Hom bounded if for every pair of objects $A,B\in\mathsf{D}$, the dimensions of $\operatorname{Hom}(A,B)$ are bounded independently of the choice of $A$ and $B$. Hom boundedness conjecture. Hom bounded triangulated categories are cone finite. This claim concerns finiteness of the possible cones of morphisms and is presented without resolution in the source.