The conjecture that convex closed semialgebraic sets are spectrahedral shadows

Let a spectrahedral shadow be the affine projection of a spectrahedron, that is, a set of the form

$$\left\{x \in {\mathbb R}^n: F_0 + \sum_{k=1}^n x_k F_k + \sum_{l=1}^p u_l G_l \geq 0,\ u \in {\mathbb R}^p\right\},$$

where $F_k \in {\mathbb S}^m$ for $k=0,1,\ldots,n$ and $G_l \in {\mathbb S}^m$ for $l=1,\ldots,p$ are given matrices. The spectrahedral-shadow conjecture. Every convex closed semialgebraic set is a spectrahedral shadow. The planar case is stated as a proposition in the surrounding text, while the general assertion is posed as the question motivating the conjecture; its resolution is not established in the supplied text.