I really don't know, but logically I think you will have to manually initiate it somehow. The theory of convertibles are like options - they are a "right" not an "obligation".

Therefore I imagine it will physically function like options currently do within your brokers platform. For options, brokers usually include an "Exercise" button next to the option once its "in the money".

When the option is not "in the money" the exercise button is usually greyed out or not present....