Concurrent normals conjecture for convex bodies

Let $C$ be a convex body in $\mathbb{R}^n$ with smooth boundary. An inward normal to $\partial C$ is a line segment or ray normal to the boundary and directed inward. The concurrent normals conjecture. There exists a point $q\in C$ such that $q$ lies on at least $2n$ inward normals to $\partial C$. The conjecture is known in dimensions two, three, and four, for centrally symmetric convex bodies, and remains open in general dimensions.