Generalised t-sum conjecture for the d-dimensional rigidity matroid

Let $G=(V,E)$, $G_1=(V_1,E_1)$ and $G_2=(V_2,E_2)$ be graphs. Say that $G$ is a $t$-sum of $G_1,G_2$ along an edge $e$ when

$$G=(G_1\cup G_2)-e,\qquad G_1\cap G_2=K_t,\qquad e\in E_1\cap E_2.$$

Here ${\mathcal R}_d$ denotes the $d$-dimensional rigidity matroid, and an ${\mathcal R}_d$-circuit is a circuit of this matroid. Generalised t-sum conjecture. Suppose that $G$ is a $t$-sum of $G_1,G_2$ along an edge $e$ for some $2\leq t\leq d+1$. Then $G$ is an ${\mathcal R}_d$-circuit if and only if $G_1,G_2$ are ${\mathcal R}_d$-circuits.

The conjecture extends the corresponding result for 2-sums. It is known when $t=d+1$ and both summands are globally rigid in ${\mathbb R}^d$, but remains open in general.