The ellipsoid full-packing conjecture for closed symplectic four-manifolds

Let $(X,\omega)$ be a closed $4$-dimensional symplectic manifold. A symplectic ellipsoid is a domain $E(a,b)\subset\mathbb{C}^2$ defined by

$$E(a,b)=\left\{(z,w)\in\mathbb{C}^2\mid \left(\pi|z|^2,\pi|w|^2\right)\in\operatorname{Conv}\left\{(0,0),(a,0),(0,b)\right\}\right\}.$$

It fully packs $X$ if there is a symplectic embedding of an ellipsoid into $X$ whose image has full symplectic volume. Ellipsoid full-packing conjecture. Every closed $4$-dimensional symplectic manifold is fully packed by one ellipsoid. This would extend full packing results from rational symplectic manifolds to arbitrary closed symplectic $4$-manifolds and would allow strong packing stability for ellipsoids to imply packing stability for all closed manifolds. The statement is presented in the source as a conjecture; no resolution is given.