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Source. Karamanlis, published p. 5, Proposition 5 (canonical PDF). The corresponding arXiv v1–v3 result is Proposition 3.3.

Statement. For every almost-regular simplex $Z$ and every integer $m\ge2$, there is an $m$-regular polygonal torus containing an isometric copy of $Z$. The polygon radii may differ.

Proof. By Lemma 4, the distance array of $Z$ is realized by labeled points in a product $\Delta_1\times\cdots\times\Delta_\ell$ of regular simplices. Matching labels gives an isometry from $Z$ to those points. Write $n_a=|\Delta_a|$. For the prescribed common order $m$, Lemma 3 embeds each $\Delta_a$ into $T_{m,r_a}^{n_a}$ for a suitable $r_a>0$. Products of these maps preserve squared distances, since each product distance is the sum of the squared distances in its factors. Composing and restricting gives an embedding of $Z$ into

$$\prod_{a=1}^{\ell}T_{m,r_a}^{n_a},$$

which is $m$-regular. $\square$

Use. In Proposition 11, the common order is chosen first by an approximation. This proposition can use exactly that order for the corrective factor; a common radius is unnecessary.