Let $\Phi$ be a root system in a real vector space $V$, and let $W = W(\Phi)$ denote its Weyl group, and let $Q = Q(\Phi) \subseteq P = P(\Phi) \subseteq V$ denote the root and weight sublattices. Moreover, let $X$ denote a $W$-invariant sublattice
$$Q \subseteq X \subseteq P$$
Is is then true that $X$ determines $\Phi$ in the sense that if $\Phi' \subseteq V$ is another root system with
$$W = W(\Phi')\quad\text{and}\quad Q(\Phi') \subseteq X \subseteq P(\Phi')$$
, then $\Phi' = \Phi$?