Degree-three generic list recovery for spherical registration

Let $\mathcal{F}$ be a finite set of nonzero spherical-harmonic frequencies, let $V$ be the corresponding $\mathrm{SO}(3)$ representation, and let $\mathbb{R}[V]^{\mathrm{SO}(3)}_d$ denote its degree-$d$ invariant polynomials. Write $\operatorname{trdeg}(\mathbb{R}[V]^{\mathrm{SO}(3)})$ for the transcendence degree of the invariant ring. Spherical-registration conjecture. Generic list recovery at degree $3$ is possible if and only if

$$\dim\bigl(\mathbb{R}[V]^{\mathrm{SO}(3)}_2\bigr)+\dim\bigl(\mathbb{R}[V]^{\mathrm{SO}(3)}_3\bigr)\geq \operatorname{trdeg}\bigl(\mathbb{R}[V]^{\mathrm{SO}(3)}\bigr).$$

In particular, when $\mathcal{F}=\{1,2,\ldots,F\}$, this holds if and only if $F\geq10$. The claim extends a theorem verified for $10\leq F\leq16$, but the general assertion is open.