Tightness conjecture for the folded-form construction of minimum-density monotone 3-subwords

Let $\Sigma_s=\{0,\ldots,s-1\}$, and for variables $x_i\geq 0$ with

$$x_1+\cdots+x_{\lfloor (s-1)/2\rfloor}\leq\frac12,$$

let $h_s(x_1,\ldots,x_{\lfloor (s-1)/2\rfloor})$ be the degree-$3$ polynomial such that every folded-form $n$-word corresponding to these parameters has monotone 3-subword density $h_s(x_1,\ldots,x_{\lfloor (s-1)/2\rfloor})+o_n(1)$. Let $f(s,3)$ denote the limiting minimum density of monotone 3-subwords over $\Sigma_s$.

Folded-form tightness conjecture. The construction's upper bound is tight:

$$f(s,3)=\min_{x_i\geq 0,\;x_1+\cdots+x_{\lfloor (s-1)/2\rfloor}\leq\frac12}h_s(x_1,\ldots,x_{\lfloor (s-1)/2\rfloor}).$$

The proposition immediately before the conjecture proves only the corresponding upper bound; equality remains open in the provided text.