Displacement-energy subadditivity for plank coverings

Let $V$ be a finite-dimensional real vector space, let $K\subset V$ be a convex body, and let $B\subset V^*$ be the unit ball of the dual norm. For the canonical symplectic structure on $V\times V^*$, write $e(K\times B^\circ)$ for the displacement energy of $K\times B^\circ$. If finitely many planks cover $K$ and their relative widths have sum $w$, then the displacement-energy conjecture asserts

$$e(K\times B^\circ)\leq 2w.$$

This conjectural bound would provide a symplectic approach to Bang-type covering problems; the source does not state a resolution.