The polynomial-exponent limit conjecture for running times

Let $H$ be an infection rule and let $M_H(n)$ denote its maximum running time. Polynomial-exponent limit conjecture. The limit

$$\lim_{n\rightarrow\infty}\frac{\log M_H(n)}{\log n}$$

exists for every infection rule $H$.

The paper does not know whether maximum running times are approximately polynomial. Existence of this limit would assign every infection rule a well-defined polynomial growth exponent, including constant, logarithmic and intermediate regimes.