The conjecture on Hofer-Zehnder subadditivity for convex coverings

Let $K_1,\ldots,K_m$ be convex bodies in $T^*\mathbb R^n$. Hofer-Zehnder subadditivity conjecture. The Hofer-Zehnder capacity of their union should satisfy

$$c_{\mathrm{HZ}}\left(\bigcup_{i=1}^m K_i\right)\leq\sum_{i=1}^m c_{\mathrm{EHZ}}(K_i).$$

This is stronger than the Akopyan–Karasev–Petrov conjecture because it does not require the union itself to be convex. The source explicitly states that this stronger assertion is unknown.