The conjecture on density, one-dimensional cones, and finite semistable classes

Let $\Lambda$ be a finite-dimensional algebra and let $g$ be a g-vector. Write $[g]_{\operatorname{TF}}$ for the $\operatorname{TF}$-equivalence class of $g$, let $\operatorname{ind}(\mathbb{N}g)$ be the set of indices associated with the nonnegative integer multiples of $g$, and let $\operatorname{TF}^{\mathrm{ss}}_{\mathbb{Z}}(g)$ denote the associated set of integral semistable objects. The conjecture. The following conditions hold:

1. Rational points are dense in $[g]_{\operatorname{TF}}$.

$$\operatorname{dim}_{\mathbb{R}}\langle\operatorname{ind}(\mathbb{N}g)\rangle_{\mathbb{R}}=1 \Longleftrightarrow [g]_{\operatorname{TF}}=\mathbb{R}^{>0}g.$$

$$|\operatorname{TF}^{\mathrm{ss}}_{\mathbb{Z}}(g)|<\infty.$$

These are presented as three simpler conjectures intended to make progress toward the broader theory of cones of g-vectors. The supplied context does not state which, if any, of the three assertions is known, so their status remains open.