In all my answers so far, I have discussed only the solvability question: does a given equation have any integer solutions or not (in this case the current smallest open equation is still $y(x^3−y)=z^3+3$ of size $H=31$, see  https://mathoverflow.net/questions/400714/).

But, of course, finding one solution does not mean to "solve" an equation. In this answer, I consider a much more general problem to decide whether the solution set is finite, and if so, list all the integer solutions. In this formulation, the smallest open equations are $y(z^2-y)=x^3-2$ and $xyz=x^3+y^2+2$ of size $H=22$. Note that for these equations the solvability question has trivial "Yes" answer, but it is open whether the solution sets are finite or infinite, see https://mathoverflow.net/questions/411958/.

If the aim is to explicitly describe all solutions (in any form), then the smallest open equations are $y^2+z^2=x^3 \pm 1$ and $y(x^2-y)=z^2-1$ of size $H=17$, see https://mathoverflow.net/questions/415173 .

Finally, if we aim to describe all solutions in parametric form only, then the smallest open equations are $yz=x^3\pm 1$ and some other equations of size $H=13$, see https://mathoverflow.net/questions/414216 for more details and the full list of equations.