Tan et al.'s labeled coupon collector difference conjecture

Let $Q_I(n,k)$ and $Q_{II}(n,k)$ denote the numbers of groups required to determine all coupon labels when the label set is known in advance and when it is not, respectively. For $k=2$, let $H_n=1+1/2+\cdots+1/n$ be the $n$-th harmonic number. Tan et al.'s difference conjecture.

$$E(Q_{II}(n,2))-E(Q_I(n,2))=\frac{1}{2}n+o(n).$$

This is one of two conjectures raised for the labeled coupon collector problem; its resolution is not specified in the source.