The generalized volume-loss asymptotic conjecture for bounded domains

Let $X\subset\mathbb{R}^4$ be a bounded domain, and define

$$\mathrm{VL}_{X}(L,R):=\inf\{\operatorname{Vol}_4({E}(\Phi))\},$$

where the infimum is over symplectic embeddings $\Phi\colon RX\to\mathbb{R}^4$ with Lipschitz constant at most $L$, and

$${E}(\Phi):=\Phi^{-1}(\mathbb{R}^4\setminus Z^4(\pi)).$$

For the ball, write $\mathrm{VL}(L,R)$ for the corresponding volume loss function. Generalized volume-loss conjecture. For all bounded domains $X$, the limit

$$s_X:=\lim_{R\rightarrow\infty}\lim_{L\rightarrow\infty}\frac{\mathrm{VL}_{X}(L,R)}{\mathrm{VL}(L,R)}$$

exists and is strictly positive. This proposes a positive asymptotic comparison between the volume loss for every bounded domain and that for balls; the supplied text gives no resolution.