The smallest-eigenvalue conjecture for Hodge Laplacians on homogeneous 3-spheres

Let $g_{(a,b,c)}$ be the homogeneous metric on the Lie group $\operatorname{SU}(2)$ or $\operatorname{SO}(3)$ determined by positive parameters $a,b,c$. For the Hodge Laplacian $\Delta_1$ on $1$-forms, write $\lambda_1(\Delta_1)$ for its smallest eigenvalue. Smallest-eigenvalue conjecture. For $(\operatorname{SU}(2),g_{(a,b,c)})$,

$$\lambda_1(\Delta_1)=\min\left(\frac{4b^2c^2}{a^2},\frac{4a^2c^2}{b^2},\frac{4a^2b^2}{c^2},a^2+b^2+c^2\right).$$

For $(\operatorname{SO}(3),g_{(a,b,c)})$,

$$\lambda_1(\Delta_1)=\min\left(\frac{4b^2c^2}{a^2},\frac{4a^2c^2}{b^2},\frac{4a^2b^2}{c^2}\right).$$

The claim concerns the global minimum over all representation weights; the preceding analysis establishes it for a class of metrics by a finite computer-assisted check, while the remaining evidence described here is numerical, so the conjecture remains open on the basis of the supplied text.