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Claim. Pavel Valtr's abstract The unit-distance problem for convex sets, in Oberwolfach Report 17/2005 (workshop Discrete Geometry, 10–16 April 2005), Oberwolfach Reports 2 (2005), no. 2, pp. 985–986, defines $t_2(n)$ as the largest number of pairs at distance exactly $1$ among $n$ pairwise disjoint translates of one compact convex set in the plane, which is $h(n)$ of Problem 956. It states, citing Erdős and Pach for the upper bound, its Theorem 2, $t_2(n)=\Theta(n^{4/3})$, and its Theorem 3, the same order when the translates are also required to be centrally symmetric. That answers the question with any $c<1/3$.

The proof. The abstract says that the talk outlined the construction; the abstract itself gives none, and no published proof is known, as Nat Sothanaphan noted on the site's thread on 27 April 2026. On 28 April 2026 Terence Tao pointed on the thread to Valtr's manuscript Strictly convex norms allowing many unit distances and related touching questions (https://kam.mff.cuni.cz/~valtr/n.pdf), whose parabolic-grid construction Chojecki's note cites; Chojecki replied that the translates in that construction may overlap, so the disjoint case needs his modification. Chojecki's claim is on its own claim page.

Depends on. Erdős and Pach's upper bound.

Standing. Claimed. Oberwolfach Reports collect workshop abstracts and are not refereed, and the abstract carries no proof. The page is dated by the issue date that Crossref records for the volume.