Electric potential of an arbitrary resistor

General case: Suppose that we have a resistor which is made from material with conductivity $sigma$. We set the voltage at the ends:

I think according to the Laplace equation we have:
$$nabla^2V = 0$$ With boundary conditions: $$ text{Portions that touch the right plate:} V = 0 text{Portions that touch the left plate:} V = V_0 text{The surrounding space is nonconducting} implies text{Other portions:} mathbf{J}.mathbf{hat{n}} = 0implies mathbf{E}.mathbf{hat{n}} = 0implies frac{partial V}{partial n} = 0$$
Where $mathbf{hat{n}}$ is the unit vector normal to the surface at each point. Is this correct?

Special case: In Griffiths "introduction to electrodynamics", there is a somewhat similar problem:

He mentions that the shape of the cross section is irrelevant but it should be the same across the resistor. So why this assumption is necessary? Also is it necessary that cross sections be perpendicular to the $z$-axis? Maybe these two assumptions are just required to make sure $V(z) = frac{V_0z}{L}$ works. If that’s the case, I think we don’t need the first assumption(i.e. the cross section stays constant across the length).

Edit: So is the formulation of the problem correct in the general case? If it’s correct, can we drop the constant cross section assumption? I mean in that case we have $$nabla^2V = 0$$ With boundary conditions: $$ text{Right cross section:} V = 0 text{Left cross section:} V = V_0 text{Other portions of the surface:} frac{partial V}{partial n} = 0$$
I think also in this case $V(z) = frac{V_0z}{L}$ solves the Laplace equation and by the uniqueness theorem, it’s the only solution.