On a conjecture Regarding the uniform distribution in the generalized coupon collector problem

The Coupon Collector’s Problem (CCP) reads as follows: how many drawings are needed on average in order to complete a collection of $n$ types of coupons, if at each step a single coupon is drawn uniformly randomly with replacement, independently of all the other drawings? This problem was introduced by De-Moivre over 300 years ago. We will discuss about a generalization of the problem, where instead of drawing a single coupon each time we draw a ``package” of coupons of size $s$ and ask the following question: how does the distribution over the collection of possible .packages affect the expected number of drawings needed to complete a collection?