Profile conjecture for $S_{2,1}$-good graph sequences

Let $S_{2,1}$ be the relevant colored graph, and let an $S_{2,1}$-good sequence $(G_n)$ have limiting edge density $\beta$ and profile $\rho(S_{2,1},(G_n))$. Define the construction densities $c(\beta)$, $cc(\beta)$, $s(\beta)$, $\ell(\beta)$, and $r(\beta)$ as in the preceding constructions. Let $x\in(0,1)$ solve $cc(x)=c(x)$, equivalently

$$16x^3-40x^2+41x-16=0,$$

and let $y\in(0,1/4)$ be a root of the degree-six polynomial determining when the optimum in $s(\beta)$ occurs at $x=0$. Profile conjecture. For every $S_{2,1}$-good sequence $(G_n)$,

$$\left.\begin{array}{ll}cc(\beta)&\text{if }\beta\in[0,x]\c(\beta)&\text{if }\beta\in[x,1]\end{array}\right\}\leq\rho(S_{2,1},(G_n))\leq\begin{cases}s(\beta)&\text{if }\beta\in[0,y]\c(\beta)&\text{if }\beta\in[y,1/4]\ell(\beta)&\text{if }\beta\in[1/4,1/2]\r(\beta)&\text{if }\beta\in[1/2,1].\end{cases}$$

This conjecture seeks the profile of $S_{2,1}$ across edge densities by comparing explicit construction lower bounds with upper bounds from the listed constructions; the exact thresholds and the sharpness of the upper bounds remain open.