Dixmier's degree-sequence conjecture for hsops of binary-form invariants

Let $I$ be the ring of invariants of binary forms of degree $n$ under the action of $\mathrm{SL}(2,\mathbf{C})$. A homogeneous system of parameters (hsop) is an algebraically independent set of homogeneous elements of $I$ such that $I$ is module-finite over the subalgebra they generate; it has $n-2$ elements. The notation $4,6,8,\ldots,2n-2$ and similar expressions denotes the listed degree sequence of an hsop. Dixmier's conjecture. (i) If $n$ is odd and $n\ge 15$, then $4,6,8,\ldots,2n-2$ is the sequence of degrees of an hsop. (ii) If $n\equiv 2\pmod 4$ and $n\ge 18$, then $2,4,5,6,6,7,8,9,\ldots,n-1$ is the sequence of degrees of an hsop. (iii) If $n\equiv 0\pmod 4$, then $2,3,4,\ldots,n-1$ is the sequence of degrees of an hsop. These proposed explicit degree sequences test the sharpness and consequences of the divisibility restrictions for systems of parameters in invariant rings of binary forms. The source attributes the statement to Dixmier; no resolution is supplied in the provided text.