The conjectured formula for the cubic supremum at large cutoff

Let $M_3(\lambda,c_0)$ be the cubic supremum, and let $c_1(\lambda)$ be a threshold such that $M_3(\lambda,c_0)=M_3(\lambda,c_1)$ for all $c_0\geq c_1(\lambda)$. For $\lambda\in[1,2]$, let $\nu_1\approx0.4501$, $\nu_2\approx1.764\times10^{-3}$, and $\nu_3\approx1.842$. Large-cutoff cubic-supremum conjecture. For all $c_0\geq c_1(\lambda)$ and $\lambda\in[1,2]$,

$$M_3(\lambda,c_0)=M_3(\lambda,c_1)=\frac{\nu_1}{\lambda},$$

where $c_1(\lambda)=\nu_3\lambda+\nu_2$. This conjecture records the numerical agreement observed for the threshold and the limiting supremum; the paper presents these values as numerical evidence rather than a proof.