At the beginning Bandler&Grinder proclaimed in The Structure of Magic that if a person access a visual memory they will always glance upwards.
Today the NLP folks don't make that claim anymore. Nowadays arguing against the orginial claim is like arguing against outdated evolutionary evidence in South Korean textbooks.
Instead the page you quoted says:
"It would be lovely if we could take this map and know instantly what a person is doing internally. But people differ and not everyone will have the same pattern. Treat this diagram as a starting point only - and use your NLP observation skills to establish what they do."
As far as the truth of the original claim goes there's a paper called NLP Eye Accessing Cues: Uncovering the Myth by Kevin Hogan. In it Kevin shows that the way Kevin understood the original claim, the original claim is false.
The NLP community believes that Kevin misunderstood the claim. According to them the sentences that Kevin used in his experiments wouldn't cause people to reliably think of a visual or auditory stimulus.
The claim that's currently believed by the NLP community is hard to test in a scientific setting.
- You need objective knowledge about whether someone thinks of an image, a sound or a feeling.
- You need subjects who are trained in NLP observation skills.
- You need somehow to eliminate the possibility that your subject use other means than Eye assessing cues in figuring out what the other person thinks.
To my knowledge no study that satisfies those concerns exists today. Most people in the NLP community don't care about academia and most people in academia don't care about NLP. You need people from both settings to fulfill those 3 criteria.