The suggestion in the article to pre-register trials is a good one, but I'm extremely wary of a more general effort to "publish rejected research" because there

The suggestion in the article to pre-register trials is a good one, but I'm extremely wary of a more general effort to "publish rejected research" because there is a huge quantity of very poorly conducted research that really does not deserve publication. Most fields are already drowning in a sea of journal articles - few researchers are aware of all the published studies that might be relevant to their own work - and greatly increasing the quantity of published material will dilute the pool even further. Replicating findings should be given higher priority, pre-registering methods and analyses should be encouraged/required, but it's important to stop short of "publish all the things".