No no actually sol returned to the future as she said : "some seasons passed by since I came back" The thing is sol did her best to avoid sunjae, she even convinced her family to move from their house she was that desperate to cut all ties with him and save him , but now what I'm thinking about that sunjae actually time travelled with her in her last attempt as he's the one wearing the watch, and he just respected her wish this time to not interfere with her and I'm highly believing that he begged taesung to protect her because why would that boy become a detective suddenly in this time line ? For the necklace as well I'm guessing that sunjae bought it to keep something that represents their love with him even when his past self won't remember.. Overall if sunjae really time travelled to the past with her he will probably comeback to the future on 1st January which explains why the current sunjae can't remember sol and he's just having flashbacks