Convex-body characterization of Lorentzian projection volumes

Let $d\geq 3$, and let $(p_{ij})$ be a real symmetric $d\times d$ matrix with nonnegative off-diagonal entries and zero diagonal entries. For $i\ne j$, let $\pi_{ij}:\mathbb{R}^d\to\mathbb{R}^{d-2}$ be the coordinate projection orthogonal to the standard basis vectors $\mathbf e_i$ and $\mathbf e_j$. A matrix is Lorentzian when it has the Lorentzian property used in the paper. Projection-volume conjecture. The following conditions are equivalent: there is a convex body $A\subseteq\mathbb{R}^d$ such that

$$p_{ij}=\operatorname{vol}(\pi_{ij}(A))\qquad\text{for all }i\ne j,$$

and the matrix $(p_{ij})$ is Lorentzian. The conjecture extends the known implication from projection volumes to Lorentzian matrices in every dimension; the converse is established when $d=4$, while the general case remains open.