  Let $B^{2n}(R)$ denote the closed $2n$-dimensional symplectic ball of area
$R$, and let $\Sigma_g(L)$ be a closed symplectic surface of genus $g$ and area
$L$. We prove that there is a symplectic embedding $\bigsqcup_{i=1}^k B^4(R_i)
\times \Sigma_g (L) \overset{s}\hookrightarrow \operatorname{int}(B^4(R))\times
\Sigma_g (L)$ if and only if there exists a symplectic embedding
$\bigsqcup_{i=1}^k B^4(R_i) \overset{s}\hookrightarrow \mathrm{int}(B^4(R))$.
  This lies in contrast with the standard higher dimensional ball packing
problem $\bigsqcup\limits_{i=1}^k B^{2 n}(R_i)\overset{s}\hookrightarrow
\mathrm{int}(B^{2n}(R))$ for $n >2$, which we conjecture (based on index
behavior for pseudoholomorphic curves) is controlled entirely by Gromov's two
ball theorem and volume considerations. We also deduce analogous results for
stabilized embeddings of concave toric domains into convex domains, and we
establish a stabilized version of Gromov's two ball theorem which holds in any
dimension. Our main tools are: (i) the symplectic blowup construction along
symplectic submanifolds, (ii) an h-principle for symplectic surfaces in high
dimensional symplectic manifolds, and (iii) a stabilization result for
pseudoholomorphic holomorphic curves of genus zero.
