Finite descriptiveness conjecture for stepped surfaces of real cubic fields

Let $\bar{\bar \alpha}=\left(\alpha_{0},\alpha_{1},\alpha_{2}\right)$ be any $\mathbb{Q}$-basis of a real cubic number field. Let $\mathscr S\left(\bar{\bar \alpha}\right)$ denote the associated stepped surface, and let $Ind^{*}=\bigcup_{n=0}^{\infty}Ind^{n}$ be the set of finite words over the index set $Ind$. For a finite patch $P$ of squares, let $\Theta_{\varepsilon}$ be the generalized substitution associated with $\varepsilon\in Ind$, and let ${}_g\Psi_s$ denote the stated geometric realization map. Finite descriptiveness conjecture. The stepped surface $\mathscr S\left(\bar{\bar \alpha}\right)$ is finitely descriptive: there exist a finite word $\varepsilon_0\varepsilon_1\cdots\varepsilon_{k-1}\in Ind^{*}$, a nonempty word $\varepsilon_k\varepsilon_{k+1}\cdots\varepsilon_{k+l-1}\in Ind^{*}$ with $k\geq0$ and $l>1$, and a finite patch $P$ consisting of squares such that

$$\mathscr S\left(\bar{\bar \alpha}\right)={}_g\Psi_s\left(\Theta_{\varepsilon_0}\Theta_{\varepsilon_1}\cdots\Theta_{\varepsilon_{k-1}}\left(\lim_{n\rightarrow\infty}\left(\Theta_{\varepsilon_k}\Theta_{\varepsilon_{k+1}}\cdots\Theta_{\varepsilon_{k+l-1}}\right)^n(P)\right)\right).$$

This conjecture asserts that every stepped surface associated with a real cubic number field can be generated from a finite initial patch by a finite prefix of substitutions followed by iteration of a periodic substitution word. The paper presents it as a consequence of the preceding conjectures together with Fernique's theorem, and the claim is supported by numerical experiments; no resolution is supplied here.