I’ve actually been using whatever human conditioners I can find on clearance in liter bottles at TJMaxx (preferably with a pump dispenser) instead of horse products.
Right now it is a white bottle that is coconut something or other. So I wash the tail with whatever shampoo I have, and then use plenty of the conditioner, let it sit for a few minutes and rinse.
The next step is, once the tail is dry, to carefully comb it out with liquid coconut oil. I think this absorbs into the hair shaft better than the conditioner that is put on when the hair is already saturated with water.
If I get around to it, after the coconut oil has had time to soak in, I do use a silicone (Cowboy Magic) to seal the whole thing and keep the hair slippery so it doesn’t snag and break.
This method helped me turn an appaloosa rat-tail into an adorable little poof of a tail that went a couple inches past her hocks by the time I sold the horse.
The tip on the clarifying shampoo is a good one - I will use that on my dark chestnut girl to try for more shine.
The problem with shining products for the show ring is - they are either silicone based, or they are oil-based and attract dirt. When I was around the western pleasure and Arabian halter people, we would fan out the tail and spray Pepi just before entering the ring. That might not be practical for dressage if you’re doing multiple tests and don’t have a groom following you around. You might get away with just putting it in the top half of the tail where it is less likely to get dirty.