I want people to care while I'm alive, to me any "care" after I die would simply be out of a selfish desire to assuage guilt, or a performance to gain sympathy.
I would not be the one receiving said care, so to kill myself for that purpose-even ever so slightly-would seem pointless,
I understand the sentiment though, most people want others to give a damn, especially if they're wading through a living hell, knocking on death's door.
The thing is, if this person you speak of doesn't care now, they won't genuinely care ever, not just because you passed.
Any pain or misgivings they experience will be more in relation to themselves, not truly about you nor any affection toward you.
You see it all the time with those who attempt, their loved ones may get a jolt or a wake up call to pay attention or to get their priorities straight, but most of the time they drift backwards toward their previous nature, and the suicidal person is left to their own devices once again.