AFAIK they mostly grow from clones, and thus, would be completely unaffected by pollination. Unless I misunderstand plants, pollination impacts the seeds produced, and therefore the subsequent generation. So long as it keeps being cut and propagated, rather than grown from seed, you could be relatively certain of near identical genetics.
For example, every apple variety out there. Every potato variety out there. Every garlic variety out there. Grape varietals. All gone because their clone lines expired. Not being able to propagate clone lines is why we can't have seedless oranges or watermelon or grapes.
