Conjecture on the dimension of the earthworm's set of holes

Let $S_n$ denote the number of holes created by the earthworm after $n$ steps, and interpret the dimension of the set of holes through the growth exponent of its expected size. Dimension conjecture.

$$\liminf_{n\to\infty} \frac{\log(\operatorname{\mathbb{E}}[S_n])}{\log(n)} > \frac{3}{4}.$$

The conjecture is motivated by simulations suggesting $S_n\sim n^\alpha$ with $\alpha\approx 0.79$, and by earlier simulations indicating a hole-set dimension close to, but possibly strictly larger than, $3/2$. The paper proves a lower bound of order $n^{3/4}$ with high probability, but does not establish the conjectured strict inequality for the expected growth exponent.