One possible synonym for "more genuinely" could be "more authentically". This word conveys a sense of sincerity and honesty, suggesting that the speaker or writer is truly expressing themselves without any pretense or affectation. Another synonym could be "more honestly", which similarly emphasizes the importance of truthfulness and straightforwardness in communication. "More earnestly" could also be a good choice, as it suggests a deep level of sincerity and commitment to one's beliefs or feelings. Whatever synonym one chooses, the key is to convey a sense of genuineness and sincerity in one's communication.