Volume Maximizing Convex Shape

Let $C$ be a convex piece of paper; its boundary may be a smooth curve, or a polygon. A perimeter halving folding is a folding of $C$ obtained by identifying two points $x$ and $y$ on the boundary of $C$ that halve the perimeter, and then folding $C$ by \textquotedblleft{}gluing\textquotedblright{} $xy$ to $yx$. This always results in a unique convex shape in 3D, a polyhedron if $C$ is a convex polygon [Demaine and O'Rourke, 2007]. What unit-area shape $C$ achieves the maximum volume possible via a perimeter-halving folding?