Generalized Penrose CASTs for odd n

Let $n$ be odd with $n\geq5$, and consider the CASTs described in Theorem~. Let $\mu_{n,\lfloor n/2\rfloor}$ be the diagonal-based inflation multiplier used there, and let $M_{n,\lfloor n/2\rfloor}$ be the corresponding substitution matrix. A prototile is one tile type in the substitution system, and a substitution rule specifies its substituted patch. Generalized Penrose CAST conjecture. CASTs of this type with inflation multiplier

$$\eta=\mu_{n,\left\lfloor n/2\right\rfloor}$$

and substitution matrix

$$M_n=\left(M_{n,\left\lfloor n/2\right\rfloor}\right)^2$$

exist with exactly $\lfloor n/2\rfloor$ prototiles and exactly $\lfloor n/2\rfloor$ corresponding substitution rules for every odd $n\geq5$. The claim extends the Penrose example to all odd orders considered, with the displayed matrix and number of prototiles prescribing the common combinatorial structure; examples are reported for several small values of $n$, while existence for every odd $n\geq5$ is the conjectural part.