Finiteness conjecture for elementary wild character varieties of fixed even dimension

Let $n$ be an even integer. An elementary wild character variety is a wild character variety arising from an elementary connection, considered up to isomorphism. Finiteness conjecture. There is a finite number of isomorphism classes of elementary wild character varieties of dimension $n$. This is conditional on the expectation that the Fourier transform induces symplectic algebraic isomorphisms between the corresponding wild character varieties. The preceding finiteness theorem for locally minimal level data is intended to yield this classification consequence, but the required general Fourier-transform description of Stokes data is not yet established.