The half-integrality conjecture for peg-solitaire linear tests

Let $\mathfrak{S}$ be the set of points of a peg-solitaire board. Let $V(\mathfrak{S},\mathbb{Z})$ be the integer span of legal-move vectors and let $V^+(\mathfrak{S},\mathbb{Q})$ and $V^+(\mathfrak{S},\mathbb{Z})$ denote their non-negative rational and integer cones, respectively. Half-integrality conjecture. If $\mathfrak{S}$ has no isolated points, then

$$V(\mathfrak{S},\mathbb{Z})\bigcap V^+(\mathfrak{S},\mathbb{Q})\subset\tfrac12V^+(\mathfrak{S},\mathbb{Z}).$$

This conjecture asserts that every vector passing the integer and rational linear tests has a non-negative representation whose coefficients have denominators at most $2$.