Conjecture on seven consecutive primitive elements in finite fields

Let $q$ be a prime power and let $\mathbb{F}_q$ denote the finite field with $q$ elements. Consecutive primitive elements are consecutive field elements that are all generators of $\mathbb{F}_q^\times$. Seven-consecutive-elements conjecture. The finite field $\mathbb{F}_q$ has $7$ consecutive primitive elements when $q$ is not divisible by $2$, by $3$, or by $5$, and when $q>1037401$.

The claim is part of the paper's computationally motivated list concerning longer strings of consecutive primitive elements, and no proof or disproof is supplied in the given text.