Assume that you draw coupons uniformly at random from a collection of $n$ coupons and you want to collect $m_i$ coupons of type $i$. This is referred to as the coupon collector with quota (http://www.combinatorics.org/ojs/index.php/eljc/article/download/v15i1n31/pdf)

Assume now that you draw your coupon by batches of $k\le n$ distinct coupons and let $T_{k,\vec{m}})$ be the number of coupons that one has to buy in order to collect $m_i$ coupons of type $i$ (for each $i$).  

Intuitively (and numerically), the expectation $\mathbb{E}[T_{k,\vec{m}}]$ is decreasing in $k$ but I am not able to find a reference (or a reasonably small proof).

My questions are:

- does anyone know a reference? 
- is it still true if the batches are of random size or if the coupons have non uniform probabilities? 

Note that the $T_{k,\vec{m}}$ are not stochastically ordered. For example when $\vec{m}=(2,0)$, we have $T_{1,\vec{m}}$ and $T_{2,\vec{m}}$ have the same mean but none of them is stochastically greater than the other one.