Classification conjecture for weighted type compositions of complete simple games

A type composition is a vector $\widehat{n}=(n_1,\dots,n_t)\in\mathbb{N}_{>0}^t$ recording the numbers of agents in the types of a complete simple game. For type compositions $\widehat{n}$ and $\widehat{m}$, write $\widehat{n}\ge\widehat{m}$ when the comparison is componentwise, with the type positions understood as in the classification. The list referred to below consists of

$$(2,4), (2,2,2), (1,1,5), (1,2,3), (1,3,2), (2,1,4), (2,4,1), (1,1,1,3), (1,1,3,1), (2,1,2,1), (2,3,1,1), (1,2,2,1), (1,2,1,2), (1,1,2,2), (1,2,1,1,1), (1,1,2,1,1), (1,1,1,2,1), (1,1,1,1,2).$$

Classification conjecture. Each type composition $\widehat{n}$ is either weighted or there exists a type composition $\widehat{m}$ in this list with $\widehat{n}\ge\widehat{m}$. The conjecture proposes that these are precisely the minimal type compositions admitting non-weighted complete simple games, while all remaining type compositions are weighted.