First Steklov eigenvalue and eigenspace conjecture for cuboids

Let $0\leq a\leq b\leq c=1$, and let $\sigma_1$ denote the first Steklov eigenvalue of the cuboid. Define $\lambda_1$ and $\lambda_2$ as the parameters occurring in the separated eigenfunction described below. First Steklov eigenspace conjecture. The eigenvalue $\sigma_1$ is always due to the eigenfunction

$$\cosh(\lambda_1 x)\cosh(\lambda_2 y)\sin\left(\sqrt{\lambda_1^2+\lambda_2^2}\,z\right).$$

If $a<b<1$, this is the unique eigenfunction giving $\sigma_1$; otherwise, other eigenfunctions may give the same eigenvalue, so $\sigma_1$ becomes multiple. This conjecture concerns the structure and multiplicity of the first Steklov eigenspace on a cuboid; the supplied text gives no resolution.