Subadditivity of Hofer–Zehnder capacity for symplectic cylinders

For a convex body $S\subset\mathbb C$, a symplectic cylinder is a set $S\times\mathbb C^{n-1}\subset\mathbb C^n$ or any of its images under symplectic transformations. Let $K\subset\mathbb C^n$ be a convex body covered by finitely many symplectic cylinders $Z_i$. The symplectic-cylinder capacity conjecture. One should have

$$c_{HZ}(K)\leq\sum_i c_{HZ}(Z_i).$$

For these cylinders, the source notes that the Hofer–Zehnder capacity equals the area of the cross-section, so this would generalize the corresponding covering result for unitary cylinders; the source does not state a resolution.