Discreteness and cone finiteness for triangulated categories

Let $\mathsf{D}$ be a Hom-finite Krull--Schmidt triangulated category and let $\mathsf{H}$ be the heart of a bounded $t$-structure. The category $\mathsf{D}$ is $\mathsf{H}$-discrete when, for every function $v\colon\mathbb{Z}\to K_0(\mathsf{H})$, only finitely many objects $D\in\mathsf{D}$ satisfy $[H^i(D)]=v(i)$ for all $i\in\mathbb{Z}$. Discreteness conjectures.

1. If $\mathsf{D}$ is $\mathsf{H}$-discrete, then all bounded hearts in $\mathsf{D}$ are discrete.
2. If $\mathsf{D}$ is $\mathsf{H}$-discrete, then $(\mathsf{D},\mathsf{H}')$ is $\mathsf{H}'$-discrete for every bounded heart $\mathsf{H}'$.
3. If $\mathsf{D}$ is $\mathsf{H}$-discrete and $\mathsf{H}$ is finite, then all bounded hearts are finite.
4. $\mathsf{D}$ is $\mathsf{H}$-discrete if and only if $\mathsf{D}$ is cone finite.

These statements are expected to hold in general and are proved in the source for derived-discrete algebras. The source also notes that discreteness with respect to a bounded heart does not force Hom boundedness, so the hypotheses distinguish different finiteness properties.