Lenore can leave flowers for the Singularity which is manifested inside the Horizon. For clarity, this Singularity structure is not the physical one that exists in the world itself, though it may contain similar mysterious properties.

Please note her first trip in, she will not have any memory of why she is there, her past, or what she learned through the Thorne mingle. Therefore, if she's consciously choosing to leave the flowers based on what she's learned, it'll need to occur during a second or future trip where she'll have her memories again. You can review the "First Visit" section of the Horizon information page for more details!

The flowers will remain until they're removed by someone else, which may happen. Over time, unless Lenore is consciously remembering she left those flowers, they may fade from existence.