The all-dimensional SIP characterization

Let $G$ be a graph, let $f$ be a nonedge of $G$, and let $G \cup f$ denote the graph obtained by adding $f$. An atom of $G \cup f$ is a graph-theoretic atom containing $f$, and an $f$-preserving $d$-forbidden minor is a forbidden minor in which $f$ is not contracted. All-dimensional SIP characterization. For any dimension $d \geq 1$, a graph-nonedge pair $(G,f)$ has the $d$-SIP if and only if no atom of $G \cup f$ that contains $f$ has an $f$-preserving $d$-forbidden minor. The theorem in the paper proves this only for $d \leq 3$, so the assertion for arbitrary $d$ remains open.