Using these definitions of soft fork and hard fork:

soft fork – tightening of the consensus rules, new blocks remain valid for old nodes

hard fork – loosening of the consensus rules, new blocks might not be valid for old nodes

It seems to me that if someone tried to deploy a pure hard fork that only loosens some consensus rules but doesn't tighten others, their new network would need to have more hashpower than the original network, or the previous chain (that remains valid for their new network) would always outpace it and no actual fork would take place. In other words, for a hard fork to be successful it should also include a soft fork. Is this correct?