There are no studies with juice, as there are no scientific studies with all those herbs as well. But those herbs doesn't have studies not only about hair grow but also everything else (plus they may have serious side-effects). On the other side, pomegranate juice have been proven to give a good amount of benefits and almost no side-effects:
https://www.uclahealth.org/news/article/health-benefits-of-pomegranates-extend-throughout-the-body
Anyway, there are scientific studies about pomegranate lotions and hair grow (but this was tested on mices):
https://www.ncbi.nlm.nih.gov/pmc/articles/PMC8100084/
For scientifically-proven products on men, only Minoxidil and Finasteride/Dutasteride and (to a lesser extent) Saw Palmetto works. The last three (Finasteride/Duta/Saw) works blocking conversion from testosterone to DHT, but they also have strong side effects on men (in particular: they strongly reduce libido/sex drive and create ED).
All the foods may contains something useful for hair grow, but the amount is so small that it's almost irrelevant at all.