Uniform Markov degree conjecture for iterated toric fiber products

Let $\mathcal{A}$ be a grading matrix, and let $\mathcal{B}_1,\dots,\mathcal{B}_s$ be arbitrary $\mathcal{A}$-graded matrices. For positive integers $r_1,\dots,r_s$, write $\operatorname*{\times}_{\mathcal{A}}^{r_i}\mathcal{B}_i$ for the $r_i$-fold toric fiber product of $\mathcal{B}_i$, and let $\operatorname{mardeg}$ denote the maximum degree of a minimal Markov basis. Then there is a constant $C\in\mathbb{N}$ such that

$$\operatorname{mardeg}\left(\left(\operatorname*{\times}_{\mathcal{A}}^{r_1}\mathcal{B}_1\right)\times_{\mathcal{A}}\left(\operatorname*{\times}_{\mathcal{A}}^{r_2}\mathcal{B}_2\right)\times_{\mathcal{A}}\dots\times_{\mathcal{A}}\left(\operatorname*{\times}_{\mathcal{A}}^{r_s}\mathcal{B}_s\right)\right)\le C$$

for all $r_1,\dots,r_s\in\mathbb{N}$.

Uniform Markov degree conjecture. The maximum Markov degree of the displayed iterated toric fiber product is bounded by a constant independent of $r_1,\dots,r_s$. This general conjecture would imply uniform degree bounds for families of iterated toric fiber products, including the preceding $K_{2,N}$ case. Its status is not resolved in the supplied source.