Differentiability and origin behavior of the first Steklov invariant surface for cuboids

Let the cuboid be $[-a,a]\times[-b,b]\times[-1,1]$, with $a,b\leq 1$, and consider the surface plotting its first Steklov invariant in the first octant. Differentiability and origin conjecture. The surface is differentiable and tends to the origin in the first octant. This conjecture is suggested by analogy with the corresponding two-dimensional situation; the supplied text gives no resolution.