Conjecture on four consecutive primitive elements in finite fields

Let $q$ be a prime power and let $\mathbb{F}_q$ denote the finite field with $q$ elements. A field has consecutive primitive elements if some sequence of consecutive elements of its multiplicative group consists entirely of generators of $\mathbb{F}_q^\times$. Four-consecutive-elements conjecture. The finite field $\mathbb{F}_q$ has $4$ consecutive primitive elements except when $q$ is divisible by $2$ or by $3$, or when $q$ is one of the following:

$$5, 7, 11, 13, 17, 19, 23, 5^2, 29, 31, 41, 43, 61, 67, 71, 73, 79, 113, 11^2, 13^2, 181, 199, 337, 19^2, 397, 23^2, 571, 1093, 1381, 7^4=2401.$$

This is presented as plausible on the basis of numerical experiments up to $10^8$, but the authors explain that a complete computational verification is very difficult because of the many large cases remaining.