The lower-bound conjecture for coupon collecting with friends

Let $n$) collectors each seek a complete collection of $m$ coupon types. The process has a collection phase of $r_c$ rounds, followed by an exchanging phase of $r_e$ rounds in which collectors interact and exchange coupons.

Lower-bound conjecture. If

$$r_c=O\left(n+\frac{n\ln n}{m}\right),$$

then

$$r_e=\Omega(m\log mn)$$

is necessary to succeed with any positive constant probability.

The conjecture asserts that the currently known upper bound on the number of interactions is tight even when only a positive constant success probability is required. The motivation is that existing lower bounds account only for the difficulty of ensuring a single interaction, not for the difficulty collectors with incomplete collections face in obtaining multiple coupons during the exchanging phase.