Pinned distinct distances for most norms

A $d$-norm is a norm $\|\cdot\|$ on $\mathbb R^d$. For a finite point set $P\subseteq\mathbb R^d$ and $x\in P$, consider the distinct distances from $x$ to the other points of $P$ in this norm.

Pinned most-norms conjecture. For most $d$-norms $\|\cdot\|$, every finite point set $P\subseteq\mathbb R^d$ contains a point $x\in P$ determining $(1-o(1))|P|$ distances to the other points, where the rate of decay $o(1)$ depends only on $\|\cdot\|$.

The conjecture is proposed as an improvement of the paper's weaker pinned-distance result for typical norms. The source does not state that it has been resolved.