Sharp-threshold conjecture for reconstructibility of random jigsaws

Let $J(n,q)$ be the random $(n,q)$-jigsaw, and let reconstructibility mean that the jigsaw is uniquely determined by its deck. The critical number of colours is conjectured to be asymptotic to $n/\sqrt{e}$. Sharp-threshold conjecture. For every $\eps>0$, as $n\to\infty$,

$$\mathbb{P}\bigl(J(n,q)\text{ is reconstructible}\bigr)\to 1$$

for all $q\geq (1/\sqrt{e}+\eps)n$. The complementary $0$-statement, namely that reconstructibility has probability tending to $0$ below the critical threshold, is established in the paper for $q\leq n/\sqrt e$. The conjecture predicts the sharp transition location, while the upper statement remains open in the interval between $n/\sqrt e$ and the proved linear upper bound.