= Solution
Take a piecewise linear <path> along three coordinate directions, passing through the origin and then $(0,1,0)$:
$$
\mathbf x(t)=\begin{cases}(1-t,0,0),&0\le t\le1,\\(0,t-1,0),&1\le t\le2,\\(0,1,t-2),&2\le t\le3.\end{cases}
$$
On the first segment $G_x=y^2+z^2=0$, so its <line integral> is zero. On the second $G_y=x^2+z^2=0$, so its contribution is also zero. On the last segment $G_z=x^2+y^2=1$ and $z$ increases from zero to one. Therefore
$$
\boxed{\int\mathbf G\cdot d\mathbf x=0+0+\int_0^1 1\,dz=1\ne\frac23.}
$$
This gives the requested concrete path dependence with the same ordered endpoints.
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