Consider $N$ disjoint circles with radius $1$ packed into a larger circle $C$. Let $R$ be the smallest possible radius of $C$, allowing the best packing density.
Now take the $N$ unitary circles and try to pack them in two other ways, again searching the maximal packing density. First, let us pack them into two disjoint and equal semicircles; then, let us pack them into four disjoint and equal quarters of circle. Let $R_S$ be the smallest possible radius of the semicircles, and let $R_Q$ be the smallest possible radius of the quarters of circle, in both cases allowing the maximal packing density. Because halving a packed circle or semicircle alters the best packing disposition (hexagonal) along the division lines, we have $R_Q>R_S>R$.
I calculated the differences $(R_S-R)$ and $(R_Q-R_S)$ using the best packing values available to date for $N\leq400$. The result is shown in the following figure:
Prove or disprove that these differences converge to definite positive values when $N$ tends to $\infty$.
Edit: the plot might suggest some convergence towards positive values. I tried to calculate these values by determining the effect of division lines on packing, and then by focusing on the region of the semicircles nearest to the diameter, and on the regions of the quarter of circles nearest to the radia. Then, I calculated the difference between the uncovered areas over these regions, considering the unitary circles first in the best packing hexagonal arrangement, and then in the "squared" arrangement (that could probably be expected at the level of diameter and radii). However, I obtained inconsistent results.