Classification conjecture for rank-two elliptic quadratic operator cones

Let $Q:{\mathbb R}^n\to{\mathbb R}^m$ be an elliptic quadratic operator with $\textup{rg}Q=2$, and let ${\overline K}'_Q$ denote its associated cone. Let $C_3$ be a three-dimensional spherical cone, let $K_r$ be a miniedral cone of dimension $r$, and let the cones of type $(C_3\oplus R_1)\cap(\widetilde C_3\oplus\widetilde R_1)$ be the cones described in the preceding example. Rank-two cone classification conjecture. The cone ${\overline K}'_Q$ is a direct sum of cones of the following types: $C_3$, possibly with many copies; $K_r$, with at most one copy; and cones of type $(C_3\oplus R_1)\cap(\widetilde C_3\oplus\widetilde R_1)$. This proposes a classification of the possible cones associated with rank-two elliptic quadratic operators; the statement is presented without a proof or resolution in the supplied text.