Conjecture on infinitely many non-covering numbers

Let $\mathscr{E}$ be the set of all integers that are covering numbers, meaning integers of the form $\sigma(G)$ for some group $G$. Non-covering-number infinitude conjecture. The set

$$\mathbb{N}-\mathscr{E}$$

is infinite; equivalently, infinitely many natural numbers are not covering numbers. The paper presents this as open and notes that its proof would require further control of groups whose covering numbers occur in the relevant ranges.