Seems to be a question not addressed in the rules. Of course, what cards you have are secret until they are played, but am I entitled to know the exact card counts of my opponents and they mine? After several plays on Hilinski's website the importance of knowing the card counts of opponents in deciding to battle on or withdraw is critical. In the online game this information is always public.
It's time for baseball, people! Pitchers and catchers report soon and the national pastime is with us again!
Every game I've ever played in, players were allowed to ask opponents how many cards they have. As you say, that can be an important consideration in the auction. Allowing this to be secret would seem to lead to silly behavior like stashing your hand of cards that I'd call counter-productive to overall enjoyment. Besides, it's hidden, trackable information. I'd say it's open data, but I don't know of an official ruling.
SoRCon 11 23-25 Feb 2018 Basildon UK http://www.sorcon.co.uk
I don't know any game I'd consider the card counts secret. (Before anyone says Acquire, those aren't cards in hand. I'm open to other games I've missed.) But there's an additional reason in the case of Taj Mahal, some cards have different backs and are clearly intended to be visible. And keeping those visible and other cards hidden is really stretching it.