Did Jasper try to talk to you about choosing your own path just before meeting the matchmaker during week 3? That's the sign that you unlocked his path. You don't need to spend more than once per week in your room to get high romance with him. Other things to do is to try and invite him to your 1st event (he will decline but you get points), be friendly to your maids and offer them breakfast, and during the trial starting in week 4 you have to do two things: 1) push for why the head butler is not letting you help Imogen more instead of dropping the topic BUT 2) when you talk to Jasper in your room in week 4 DO NOT try to push for more info out of him. You also get an automatic scene where you worry about him, so choose the option where you ask him to lay down his burdens. Once you get him to sit down and talk to you, pretty much all of the choices afterwards for that conversation will get you romance points from him. After week 4, just continue to go to your room once a week and you will be in the clear!
If you are too busy with increasing your stats, you do NOT need to complete the historian chain, but doing so will definitely get you a happier ending with Jasper, so I recommend it! It also explains some of his behaviors and gives some insight into his backstory (I've been told though that we won't be able to fully find out what is going on though until the secret backgrounds are released).