Updating an Existing Property on Workflow vote

I don´t see any "apply" in your code. You set the Property, but only to the temporary context. You don´t seem to use edit/apply to actually save your new value. 

And I see a second problem. As far as I understand you want to update your history value:

You use this line to get the value:

string history = controlledItem.getProperty("_conversation_history","");

But that line to update the value:

this.setProperty("_conversation_history",newconvo);

In your context "this" represents the "Activity" item while controlledItem represent your change process. 
I am not sure if controlledItem.setProperty in combination with edit/apply would work, cause the controlled item is something generated by the helper function. But you can give it a try. 

If it doesn´t work, get the id of the controlledItem and do a direct edit of the Change process.