Conjecture on six 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 $\mathbb{F}_q^\times$. Six-consecutive-elements conjecture. The finite field $\mathbb{F}_q$ has $6$ consecutive primitive elements when $q$ is not divisible by $2$, by $3$, or by $5$, and when $q>65521$.

This and the related claims for seven and eight consecutive primitive elements are reported from computational examination rather than established proofs; the source does not provide a resolution.