The doubled-basis conjecture for peg-solitaire move spaces

Let $\mathfrak{S}$ be a peg-solitaire board, let $\mathscr{D}(\mathfrak{S})$ be its set of legal-move vectors, and let $\mathcal{B}\subset\mathscr{D}(\mathfrak{S})$ be a basis of $V(\mathfrak{S},\mathbb{Q})$. Define

$$V(\mathcal{B})=\sum_{{\mathfrak{f}}\in\mathcal{B}}\mathbb{Z}\cdot{\mathfrak{f}}.$$

Let $\mathscr{F}(\mathfrak{S},\mathbb{Z})$ denote the relevant integer lattice generated by the legal-move vectors. Doubled-basis conjecture. If $\mathfrak{S}$ has no isolated points, then

$$2\mathscr{F}(\mathfrak{S},\mathbb{Z})\subset V(\mathcal{B})=\sum_{{\mathfrak{f}}\in\mathcal{B}}\mathbb{Z}\cdot {\mathfrak{f}}.$$

This is proposed as a related, potentially easier-to-handle formulation of the half-integrality phenomenon for the linear test.