The maximal-rank conjecture for regular polytopes with automorphism group $\mathrm{A}_n$

Let $n\geq 12$. An abstract regular polytope with automorphism group $\mathrm{A}_n$ has rank $d$, where rank means the number of involutory generators in its string C-group representation. Examples of rank $\lfloor (n-1)/2\rfloor$ are known. Maximal-rank conjecture. The highest rank of an abstract regular polytope having $\mathrm{A}_n$ as automorphism group is

$$\left\lfloor \frac{n-1}{2}\right\rfloor.$$

This conjecture concerns the largest possible rank of a string C-group representation of the alternating group. The stated lower-bound examples are known, while the conjectured optimality for $n\geq 12$ remains open.