Poonen–Stoll conjecture on rational points on genus-two curves

Consider the density obtained by ordering genus-two curves of the form $y^2=f(x)$, where $f=f_6x^6+\dots+f_1x+f_0\in{\mathbb Z}[x]$, by the condition $\max\{|f_j|\}\leq N$, and letting $N$ tend to infinity. Poonen–Stoll conjecture. The density of curves of genus $2$ with rational points is $0\%$. The source contrasts this with the approximately $84$–$85\%$ density of such curves having points everywhere locally, so the conjecture asserts that the Hasse principle fails for almost all curves in this ordering.