The single-interval characterization of d-convex graph-nonedges

Let $d\geq 1$, let $G$ be a graph, and let $F$ be a set of nonedges of $G$. For each $f\in F$, write $(G\cup(F\setminus\{f\},f))$ for the graph-nonedge pair obtained by adding the other nonedges in $F$ to $G$ while retaining $f$ as the distinguished nonedge. The pair $(G,F)$ is $d$-convex when it satisfies the paper's $d$-convexity condition, and a graph-nonedge pair has the $d$-SIP when its Cayley configuration space is single interval in dimension $d$. The $d$-convexity conjecture. For any dimension $d\geq 1$ and any graph $G$ with nonedge set $F$, $(G,F)$ is $d$-convex if and only if, for every $f\in F$, $(G\cup(F\setminus\{f\},f))$ has the $d$-SIP. This would give an equivalent single-interval characterization of $d$-convexity for arbitrary nonedge sets, extending the stated low-dimensional relationship between convexity and the $d$-SIP. The claim is presented as a conjecture in the paper, and no resolution is supplied here.