Complete bullshit. Akiha is a COMPLETE BITCH, she would just say he doesn't have the right to end his life because she never gave permission. Not to mention her feelings for him are EXACTLY like that of Gollum to the one ring(an obsession of a POSSESSION).
Shiki doesn't give a shit about making Akiha "sad", any time he thought of that it was nothing more than an after thought. He'd put all priority on his ACTUAL word, not some after thought he made when he had nothing to do.