Clean lex tuple conjecture for monic Weierstrass hypersurfaces

Let $g$ and $h$ be the base polynomials in the monic Weierstrass family of hypersurfaces

$$z^3+g(x,y,w)+z^2h(x,y,w),$$

and let the clean lex tuple be

$$\mathcal R_{\mathrm{lex}}=(c_1,c_2,c_3,c_4,c_5).$$

A bounded-delay descent condition means that improvements in this tuple occur within a fixed finite window of canonical steps. Clean lex tuple conjecture. On this monic Weierstrass family, $\mathcal R_{\mathrm{lex}}$ satisfies the bounded-delay descent condition for some universal window $m$; empirically, $m=5$. The claim is motivated by the focused benchmark, while the unrestricted family has a computational counterexample for $m=10$, so the required restrictions on $g$ and $h$ and the existence of a universal window remain open.