Sufficiency conjecture for isostatic frameworks with $C_{2v}$ or $C_{3v}$ symmetry

Let $G=\{J,B\}$ be the connectivity of a framework, with $j=|J|$ joints and $b=|B|$ bars, and let $p$ be a plane configuration generic with symmetry group ${\cal G}$. Write $j_c$ for the number of joints at the center, $b_2$ for the number of bars fixed by the half-turn, and $b_{\sigma}$ for the number of bars fixed by a mirror $\sigma$. For any non-empty set of bars $B^{*}$ contacting only the joints in $J^{*}$, write $b^{*}=|B^{*}|$ and $j^{*}=|J^{*}|$. Sufficiency conjecture. If $G(p)$ is realized with symmetry group ${\cal G}=C_{2v}$ or ${\cal G}=C_{3v}$, then the following necessary conditions are also sufficient for $G(p)$ to be isostatic:

$$b=2j-3,$$ $$b^{*}\leq 2j^{*}-3$$

for every non-empty set of bars $B^{*}$, together with $b_2=1$ and $b_{\sigma}=1$ for each mirror when ${\cal G}=C_{2v}$, and $j_c=0$ and $b_{\sigma}=1$ for each mirror when ${\cal G}=C_{3v}$. This extends the sufficiency result already established for the plane symmetry groups $C_s$, $C_2$, and $C_3$; the remaining issue is whether these conditions suffice for the two groups $C_{2v}$ and $C_{3v}$, beyond their necessity.