The sufficient condition for finite generation of simple irrational polyhedral cones

Let $C\subset\mathbb{R}^n$ be a simple polyhedral cone with extreme rays $[u_1],\dots,[u_n]$, where $[u_1],\dots,[u_k]$ are rational.

Finite-generation conjecture. The cone $C$ is $(R,G)$-finitely generated if and only if there exists $A\in\mathbb{Q}^{n\times n}$ and $\lambda_1,\dots,\lambda_n\in\mathbb{R}_{>0}$ such that $Au_i=\lambda_i u_i$ for $i=1,\dots,n$ and $\lambda_{k+1},\dots,\lambda_n$ are distinct.

The sufficient condition is known by Theorem; the conjecture asserts that it is also necessary. The surrounding discussion notes that the necessity is established in dimension three, while higher-dimensional examples with repeated eigenvalues show that the general question remains open.