Unique Apéry expansions and Cohen–Macaulay tangent cones for the semigroups Γ_m

Let $\Gamma_m$ be the numerical semigroup generated by $s_n=\frac{n}{2}[2a+(n-1)d]$ for $1\leq n\leq m$, where $a,d>0$ and $\gcd(a,d)=1$. Its Apéry set with respect to $a$ is

$$\operatorname{Ap}(\Gamma_m,a)=\{s\in\Gamma_m\mid s-a\notin\Gamma_m\}.$$

Unique-expansion conjecture. The elements of $\operatorname{Ap}(\Gamma_m,a)$ have unique expressions. Hence the tangent cone of $k[[\Gamma_m]]$ is Cohen–Macaulay. This would extend the paper’s verified results for the family under study and would provide Cohen–Macaulayness of the associated tangent cones for all relevant $m$; the source gives no resolution of the conjecture.