The one-point deletion conjecture for planar point configurations

Let $P,Q\in\mathbb{R}^2$ be two $n$-point configurations with equal distance distributions $D_P=D_Q$. Assume the points are labelled appropriately, and suppose there do not exist $A\in O(2)$ and $\vec{t}\in\mathbb{R}^2$ such that

$$Q=AP+\vec{t}.$$

One-point deletion conjecture. There exist $p_i\in P$ and $q_i\in Q$, together with some $A\in O(2)$ and $\vec{t}\in\mathbb{R}^2$, such that

$$Q\backslash q_i=A(P\backslash p_i)+\vec{t}.$$

The conjecture asserts that two noncongruent planar configurations with the same distribution of pairwise distances can be made congruent by deleting one point from each configuration. The surrounding examples motivate the claim, but the supplied text gives no resolution.