This is indeed a good question. In order to come up with an answer, let us imagine the following. If you take hexan-1-ol, cyclohexanol, and 2,2-dimethylbutan-2-ol and treated each with the same mixture of phosphoric acid and a small amount of sulfuric acid. Let us further assume that kinetics of the tertiary alcohol are unimolecular (E1) and the primary alcohol are bimolecular (E2). What are the kinetics of the secondary alcohol and how would you know for sure?
I don't know that you could know for sure. If you had 2- and 3-methylcyclohexanols and treated them as before, you could begin to infer whether the reactions had carbocation intermediates from the product distribution. However, I don't know that you could exclude E1 and E2 mechanisms competing in the reactions leading to the products.
In Williamson's Macroscale and Microscale Organic Experiments, the reaction mechanism is shown as an E1 reaction. I don't know what proof was used for it though.