Kakeya set with high Turing join

Construct a Kakeya set A in R-squared such that the Turing join of the coordinates of every point in A computes the halting problem.

Background

The problem imposes a computability-theoretic lower bound on every point of a Kakeya set, requiring the coordinate join to compute the halting problem.

References

Is there a Kakeya set $A\subseteq\mathbb{R}2$ such that $x\oplus y\geq_T\emptyset'$ for every $(x,y)\in A$?

— Open Problems in Mathematical Logic  (2608.26628 - Barmpalias et al., 27 Aug 2026) in Section 4