Conjecture on five 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 sequence of consecutive primitive elements is a sequence of consecutive elements of $\mathbb{F}_q$ whose members all generate $\mathbb{F}_q^\times$. Five-consecutive-elements conjecture. The finite field $\mathbb{F}_q$ has $5$ consecutive primitive elements except when $q$ is divisible by $2$ or by $3$, or when $q$ is one of the following:

$$5, 7, 11, 13, 17, 19, 23, 5^2, 29, 31, 37, 41, 43, 47, 7^2, 61, 67, 71, 73, 79, 101, 109, 113, 11^2, 5^3, 127, 131, 139, 151, 157, 163, 13^2, 181, 193, 199, 211, 229, 241, 271, 277, 281, 17^2, 307, 313, 331, 337, 19^2, 379, 397, 433, 439, 461, 463, 23^2, 547, 571, 577, 601, 613, 5^4, 631, 691, 751, 757, 29^2, 31^2, 1009, 1021, 1033, 1051, 1093, 1201, 1297, 1321, 1381, 1453, 1471, 1489, 1531, 1597, 1621, 1723, 1741, 1831, 43^2, 1861, 1933, 2017, 2161, 2221, 2311, 2341, 7^4, 3061, 59^2, 3541, 3571, 61^2, 4201, 4561, 4789, 4831, 71^2, 5281, 5881, 89^2, 8821, 9091, 9241, 113^2, 5^6=15625.$$

The conjecture is based on numerical experiments up to $10^8$; the source notes that settling it computationally is difficult because many very large cases remain to be tested.