Dixmier's HSOP conjecture for binary form invariants

Let $S_d$ be the space of binary forms of degree $d$, let ${\rm Inv}_d$ be its ring of $SL_2$-invariants, and, for $d=2k$ with $k$ even, let $\mathscr{H}_{n,p}(F)$ denote the coefficient of $\lambda^p$ in the characteristic polynomial of the transvectant map $\mathcal{L}_n^F:S_n\to S_n$, $G\mapsto(F,G)_k$. Dixmier's conjecture. When $d$ is divisible by $4$, the invariants

$$\mathscr{H}_{d-2,2}(F),\mathscr{H}_{d-2,3}(F),\ldots,\mathscr{H}_{d-2,d-1}(F)$$

form a homogeneous system of parameters for ${\rm Inv}_d$. This conjecture proposes an explicit parameter system for the invariant ring of binary forms; the source presents it as Dixmier's conjecture and does not state a resolution.