Finiteness and strict growth conjecture for compact packings with n radii

For each $n\in\mathbb{N}$, let $\Pi_n$ denote the set of compact packings of the plane with $n$ distinct circle radii, represented by radius ratios $(r_{n-1},r_{n-2},\ldots,r_1)$. The construction obtained by filling interstitial gaps of a compact $(n-1)$-packing corresponds to the truncation $(r_{n-2},\ldots,r_1)\in\Pi_{n-1}$. Finiteness and strict growth conjecture. For every $n\in\mathbb{N}$, the set $\Pi_n$ is finite and the sequence $(|\Pi_n|)_{n\in\mathbb{N}}$ is strictly increasing. Furthermore, for every $n\in\mathbb{N}$, there exists an element $(r_{n-1},r_{n-2},\ldots,r_1)\in\Pi_n$ with $(r_{n-2},\ldots,r_1)\notin\Pi_{n-1}$; equivalently, not every compact $n$-packing arises from filling interstitial gaps of a compact $(n-1)$-packing. The conjecture extends the observed finiteness of $\Pi_2$ and $\Pi_3$ and predicts genuinely new compact packings at every number of radii, rather than only those obtained by recursively filling gaps.