Extension of norms from subsets of integer vectors

Let $G\subseteq\mathbb Z^3\setminus\{0\}$, and suppose that a function $\lVert\cdot\rVert:G\to\mathbb R_{\geq 0}$ satisfies $\lVert g\rVert>0$ for every $g\in G$ and, whenever

$$g=\sum_{h\in G}\lambda_hh$$

with finitely many nonzero $\lambda_h\in\mathbb R$, satisfies

$$\lVert g\rVert\leq\sum_{h\in G}|\lambda_h|\,\lVert h\rVert.$$

Norm-extension conjecture. Then $\lVert\cdot\rVert$ is the restriction to $G$ of a norm on $\mathbb Z^3$.

The conjecture concerns extending prescribed values on a subset of integer vectors to a norm-induced metric. The source introduces it as a proposed generalization beyond the fixed $1$-norm setting; no resolution is supplied.