$p_k(M,\omega)=1$ we say that $(M,\omega)$ admits a full packing by $k$ balls, otherwise we say that there is a packing obstruction. An essentially equivalent invariant is the generalized $k^{\text{th}}$ Gromov width ${\mathsf{w}}_k(M,\omega)$ defined by setting $${\mathsf{w}}_k(M,\omega) = \sup_{c>0} \{ c ~|~\sqcup_k B(c) \text{~embeds symplectically into~} (M,\omega) \}$$ For a compact manifold of dimension $2n$ the width ${\mathsf{w}}_{k}$ is thus bounded by $$0<{\mathsf{w}}_k(M^{2n},\omega)\leq c_{\operatorname{vol}}(M^{2n},\omega):= \sqrt{\frac{n!\operatorname{vol}\left(M^{2n},\omega\right)}{k}}$$ Although no general tools are known to compute those invariants for arbitrary symplectic manifolds, some results can be derived from complex algebraic geometry. For instance, in [@MP], D. McDuff and L. Polterovich computed $p_{k}({\mathbb{CP}}^{2})$, for $k\leq 9$. They also proved that $p_{k}({\mathbb{CP}}^{n}) = 1$ whenever $k=p^{n}$ and that $\lim_{k\to\infty} p_k(M,\omega) = 1$ for any compact symplectic manifold. In view of that later result, it is natural to ask whether the sequence $p_k(M,\omega)$ is eventually stable, that is, whether there is a number $N_{{\rm stab}}(M,\omega)$ such that $p_k(M,\omega)= 1$ for all $k \geq N_{{\rm stab}}(M,\omega)$. To date, this remains an interesting open question (see [@CHLS] and [@Bi-SPAG] for a complete discussion). The only general result in that regard is due to P. Biran ([@Bi], [@Bi2]) who settled this question positively for all closed symplectic $4$-manifolds whose symplectic 