The convolution-ratio conjecture for probability density functions

Let $f$ be a probability density function supported on $[-\frac14,\frac14]]$. The convolution $f\ast f$ has supremum norm $\|f\ast f\|_\infty$ and $L^2$ norm $\|f\ast f\|_2$. Convolution-ratio conjecture. If $f$ is a pdf supported on $[-\frac14,\frac14]$, then

$$\frac{\|f\ast f\|_\infty}{\|f\ast f\|_2^2} \geq \frac{\pi/2}{\log 4},$$

with equality only if either $f(x)$ or $f(-x)$ equals $\sqrt{\frac2{4x+1}}$ on the interval $\lvert x\rvert\leq\frac14$. This would improve the lower bound for autoconvolutions discussed in the paper, and the source presents it as a belief rather than reporting a resolution.