Classification conjecture for double bubbles in radial log-convex density

Let $N\geq 1$, and let $\mathbb{R}^N$ have a smooth, radial, log-convex density. For prescribed volumes $V_1$ and $V_2$, consider a perimeter-minimizing double bubble. A bubble inside a bubble is the configuration in which one bubble is enclosed by the other, while a standard double bubble is the usual adjacent two-bubble configuration.

Higher-dimensional double-bubble classification conjecture. A perimeter-minimizing double bubble is either

$$\text{(i) the bubble inside a bubble, for example when }V_1\text{ is small and }V_2\text{ is large,}$$

or

$$\text{(ii) the standard double bubble, for example when }V_2\text{ is close to }V_1.$$

Numerical experiments in dimensions two and three motivate this classification, as an analogue of the one-dimensional double-interval/triple-interval alternatives. The conjecture does not address uniqueness or the transition between the two configurations.