Baranyai's partition characterisation of (3,6)-tight graphs

Let $G=(V,E)$ be a graph with at least $4$ vertices. A graph is $(3,6)$-tight when $|E|=3|V|-6$ and every vertex subset $U$ with at least two vertices spans at most $3|U|-6$ edges. For an edge $e$, write $(V,S)/e$ for the graph obtained by contracting $e$. A graph $H$ is $(2,3)$-tight when $|E(H)|=2|V(H)|-3$ and every vertex subset $U$ with at least two vertices spans at most $2|U|-3$ edges.

The partition characterisation conjecture. The graph $G$ is $(3,6)$-tight if and only if, for every edge $e\in E$, there exists a partition $(S_1,S_2,S_3)$ of $E$ such that

$$|S_i|=|V|-i\quad\text{for each }i\in\{1,2,3\},$$

$e\in S_1$, and each of the graphs

$$(V,S_1\cup S_2),\qquad (V,S_1\cup S_3)/e,\qquad (V,S_2\cup S_3\cup\{e\})/e$$

is $(2,3)$-tight.

This conjecture is proposed as a correction to a claimed combinatorial characterisation of minimal $3$-rigidity. The paper presents a counterexample to the earlier characterisation and suggests that the stated partition condition may provide the appropriate characterisation; its resolution is not established here.