The question is in the title. The motivation comes from trying solving Diophantine equations in order, see Can you solve the listed smallest open Diophantine equations? . Because there is an algorithm for solving all equations of genus $g\leq 1$, it is natural to search for the smallest/simplest genus $2$ equations that are difficult to solve. All genus $2$ equations are hyperelliptic, that is, can be reduced to the form $y^2=P(x)$ after rational change of variables. Then Magma code
P := PolynomialRing(Rationals()); C := HyperellipticCurve(P(x)); J := Jacobian(C); RankBounds(J);
returns the upper and lower bounds for the rank $r$ of the Jacobian. If $r\leq 1$, then $r<g$, and all rational solutions to $y^2=P(x)$ can be computed by Chabauty command implemented in Magma.
The smallest genus $2$ equation not solvable by this method is the equation $x^4-x^2+xy+y^3=0$, which transforms into the equation in the title. In this case, $r=2=g$, and Chabauty command does not work.
