Compact-support extremizer conjecture for autocorrelation inequalities

For $g\in L^1(\mathbb{R})$ with $g\geq 0$ and compact $I\subset\mathbb{R}$, let $\mathcal{L}_g(I)$ be the class of nonnegative $L^1$ functions $f$ on $\mathbb{R}$ such that $f\leq g$ outside $I$. Consider the extremal problems represented by the quantities in and. Compact-support extremizer conjecture. There exist extremizers for both problems, and the extremizers can be chosen to have compact support. The existence of compactly supported extremizers would make the restrictions to almost-compactly supported function classes largely superfluous; the paper proves existence in a larger class of positive measures for one related problem, but the asserted $L^1$ extremizer statement remains open.