The interval conjecture for region counts of circles with pairwise intersections

Let $C_n$ be the set of numbers of regions formed by $n$ circles in the plane, in an arrangement not in general position, such that every two circles have at least one common point. Let $L_n$ be the set of numbers of regions formed by $n$ distinct lines in the plane in an arrangement not in general position. The maximal elements of $L_n$ and $C_n$ are respectively $\frac{n(n-1)}{2}+1$ and $n(n-1)+2$. Interval conjecture. The set $C_n$ contains every integer in the interval from $\frac{n(n-1)}{2}+1$ through $n(n-1)+2$. This would determine a complete consecutive range of attainable region counts for the circle arrangements under consideration, extending the relation between the possible counts for circles and lines; the supplied text gives no resolution.