Disclaimer: Upon recommendation (in the comments), this post is just to close off the question, since it has been completely solvedanswered in the comments.
Indeed, as observed by user @Terry Tao in the comments, the question has a simple affirmative answer via the Riez rearrangment inequality.
Using this machinarymachinery, a more general question in $n$ dimensions is answered here https://mathoverflow.net/a/415754/78539, and the answer to my original question (on the real line) follows as an immediate corollary.