Hilbert's conjectured resolvent degrees for polynomials of degrees six, seven, and eight

Let $\operatorname{rd}_{\mathbb C}(n)$ denote the resolvent degree of the general polynomial of degree $n$ over $\mathbb C$. It is known that $\operatorname{rd}_{\mathbb C}(n)=1$ for $n\leqslant 5$.

Hilbert's conjecture.

$$\operatorname{rd}_{\mathbb C}(6)=2,\qquad \operatorname{rd}_{\mathbb C}(7)=3,\qquad \operatorname{rd}_{\mathbb C}(8)=4.$$

These values were conjectured by Hilbert and concern the first degrees for which the resolvent degree is not known. The surrounding discussion emphasizes that even whether $\operatorname{rd}_{\mathbb C}(n)>1$ for some $n$ remains unresolved; no resolution of these three specific values is supplied here.