It turns out the answer is “no”.
First, keep in mind every part of our genome coalesces at a different time. First off we need to define coalescence using TMR4A for Autosomal, TMR3A for X, and TMR1A for mitochondria and Y. Because of recombination X Autosomal actually Correspondens to greater than tens of thousands of measurements.
What we know is that:
Time of Bottleneck > Maximum(Time to Coalescents)
Here, the “greater than” sign means “more ancient than”. There is a catch that helps you. Practically speaking we can’t directly measure Coalescence time, but have to estimate it based on mutational distance. For this reason we expect some of our estimated Coalescence times to be too high, in error. Therefore it is valid in this context to relax this constraint to:
Time of Bottleneck > Median(Time to Coalescents)
If the bottleneck is more ancient than this median time, we don’t expect to see any impact on the data. It could be there, but we dont know from genetic evidence.
No contradiction unless you want a bottleneck more recent than this bound.