Linear lower-bound conjecture for distinct dot products of planar point sets

Let $\mathcal{P}=(P_1,P_2,P_3,\ldots)$ be a sequence of point configurations, where each $P_n$ is a set of $n$ distinct points $\{p_1,\dots,p_n\}$ in $\mathbb{R}^2$. Define the set of dot products by

$$D(P_n)=\{p_i\cdot p_j\mid p_i,p_j\in P_n\}.$$

Linear distinct-dot-products conjecture. For every such sequence,

$$|D(P_n)|\gtrsim n.$$

The conjecture asserts a linear lower bound for the number of distinct dot products determined by an arbitrary planar configuration. The source gives no resolution evidence, so its status is recorded as open; the abstract notes that the best known lower bound is sublinear while known constructions scale linearly.