Pinned dot product measure conjecture

Let $A\subset\mathbb{R}^n$ be a Borel set, and for $a,x\in\mathbb{R}^n$ define the pinned dot product set

$$\Pi_x^a(A)=\{\alpha\in\mathbb{R}:(a-x)\cdot y=\alpha\text{ for some }y\in A\}.$$

In particular, $\Pi_0^a(A)$ is the dot product set pinned at $a$. Pinned dot product measure conjecture. If

$$\dim(A)>\frac{n}{2},$$

then there exists a pin $a\in A$ such that

$$\mathcal H^1\bigl(\Pi_0^a(A)\bigr)>0.$$

This conjecture is motivated by the Falconer distance conjecture and by the paper's translated pinned dot product results, which attain the same dimensional threshold. Its resolution is not stated in the supplied text.