Now deathsinger checkpoint is technically considered as a chest, meaning that it can be obtained once a week like the loot chest in VoG and CE. If you complete the normal mode CE before, you will not get any reward in deathsinger checkpoint in hard mode CE this week.
Bungie, you know what you have to do.
Edit: I noticed there are four columns in CE reward advisor now, I cannot confirm it just yet.
Edit: I have tested on two character, I completed one normal on my warlock and completed hard on my titan. the advisor show three columns completed on my warlock, and full clear on my titan. And yet the rewards are the same.