Supermodularity conjecture for a convex body and compact summands

Let $n\geq 2$. For a convex body $A$ and compact sets $B,C\subset\mathbb{R}^n$, consider their Minkowski sums. Supermodularity conjecture.

$$|A+B+C|+|A|\geq |A+B|+|A+C|.$$

This extends the known one-dimensional inequality involving $|\operatorname{conv}(A)|$ and asks whether convexity of $A$ restores supermodularity in higher dimensions; the conjecture is open for $n\geq2$.