Kiraly–Tanigawa's body-pin rigidity characterization conjecture

Let $H=(W,E)$ be a multigraph, and let $G_H$ be the body-pin graph obtained by replacing each $w\in W$ with a body and identifying distinct pairs of body vertices according to the edges of $H$. For disjoint subsets $X,Y\subseteq W$, let $d_H(X,Y)$ be the number of edges between them and define

$$\ell_H(X,Y)= \begin{cases} 6 & \text{if }d_H(X,Y)\geq 3,\\ 5 & \text{if }d_H(X,Y)=2,\\ 3 & \text{if }d_H(X,Y)=1,\\ 0 & \text{if }d_H(X,Y)=0. \end{cases}$$

Kiraly–Tanigawa's body-pin conjecture. The graph $G_H$ is rigid in $\mathbb{R}^3$ if and only if, for every partition $\mathcal{P}=\{P_1,P_2,\dots,P_t\}$ of $W$,

$$\sum_{1\leq i<j\leq t}\ell_H(P_i,P_j)\geq 6(|\mathcal{P}|-1).$$

This is a conjectured characterization of rigidity for body-pin graphs in three dimensions. The source attributes it to Kiraly and Tanigawa and does not report a resolution.