Conjecture on higher Ekeland-Hofer capacities of symplectic p-products

Let $X_1\subset\mathbb R^{2n}$ and $X_2\subset\mathbb R^{2m}$ be star-shaped domains, let $c^k_{\rm EH}$ denote the Ekeland-Hofer capacities, and let $X_1\times_pX_2$ be their symplectic $p$-product. Set $c^0_{\rm EH}=0$. For $1\leq p$,

$$c^k_{\rm EH}(X_1\times_pX_2)= \begin{cases} \displaystyle\min_{i+j=k}\left[c^i_{\rm EH}(X_1)^{\frac p{p-2}}+c^j_{\rm EH}(X_2)^{\frac p{p-2}}\right]^{\frac{p-2}{p}},&p\geq2,\\ \displaystyle\max_{\substack{i+j=k+1\i,j\ne0}}\left[c^i_{\rm EH}(X_1)^{\frac p{p-2}}+c^j_{\rm EH}(X_2)^{\frac p{p-2}}\right]^{\frac{p-2}{p}},&1\leq p\leq2. \end{cases}$$

The higher-capacity p-product conjecture. The displayed formula should hold for every such pair of star-shaped domains and every admissible $p$. The conjecture generalizes the known Cartesian-product formula for Ekeland-Hofer capacities. The source does not state a resolution status.