Einstein problem (aperiodic monotile)

Does there exist a single connected closed topological disk $T \subset \mathbb{R}^2$ (an "einstein", from the German ein Stein, "one stone") such that congruent copies of $T$ can tile the entire Euclidean plane $\mathbb{R}^2$ without overlaps or gaps, yet every such tiling is non-periodic (invariant under no nonzero translation)?

Solved in 2023 by David Smith, Joseph Samuel Myers, Craig S. Kaplan, and Chaim Goodman-Strauss. In March 2023 they announced the "hat", a 13-sided polykite made of eight kites from the Laves $[3.4.6.4]$ tiling, and proved via hierarchical substitution metatiles and a computer-assisted case analysis of surrounding clusters that it tiles $\mathbb{R}^2$ only non-periodically (using both unreflected and reflected copies). Two months later, in May 2023, the same four authors introduced the "Spectre", a 14-sided tile with gently curved or notched edges that is a strictly chiral aperiodic monotile: even when reflections are permitted, the Spectre tiles the plane only using pure translations and rotations, and every tiling is non-periodic.