The conjecture that the quadratic supremum equals the sharp lower bound

Let $G$ be the energy function and let $M_2(\lambda,c_0)$ denote the corresponding supremum, with parameters $\lambda>0$ and $c_0>0$. Quadratic-supremum conjecture. For all $\lambda>0$ and $c_0>0$,

$$M_2(\lambda,c_0)=\sqrt{2}.$$

The claim is motivated by numerical calculations showing that $M_2(\lambda,0)$ differs from $\sqrt{2}$ by approximately $2.7\times 10^{-9}$ for $\lambda\in[1,2]$, while the established bounds give $M_2(\lambda,0)\geq M_2(\lambda,c_0)\geq\sqrt{2}$.