Arnold's filtered invariants conjecture for spherical curves

Let $J^+ + 2 St$ be the Arnold invariant of spherical curves, and let $C$ be an immersion $S^1 \to S^2$. For each $i\geq 0$, let $E_i$ be an invariant of such immersions.

Arnold's conjecture. There exists a sequence of invariants $\{E_i\}_{i\geq 0}$ with $E_0=J^+ + 2 St$ such that

$$\{C\mid E_0(C)=0\}\supset \{C\mid E_1(C)=0\}\supset \{C\mid E_2(C)=0\}\supset\cdots.$$

This conjecture seeks a filtered sequence of invariants for plane or spherical curves, analogous to the filtration by Vassiliev-type invariants and extending Arnold's invariant. The supplied source does not state whether the conjecture has been resolved.