Autoconvolution norm-ratio conjecture for probability density functions

Let $f$ be a probability density function supported on $[-\frac14,\frac14]$. The norms $\|f\ast f\|_\infty$ and $\|f\ast f\|_2$ are defined using the autoconvolution $f\ast f$. Autoconvolution norm-ratio conjecture.

$$\frac{\|f\ast f\|_\infty}{\|f\ast f\|_2^2} \geq \frac{\pi}{\log 16},$$

with equality only if either $f(x)$ or $f(-x)$ equals $\sqrt{\frac2{4x+1}}$ on the interval $|x|\leq\frac14$.

This conjecture would strengthen the lower bounds obtained for autoconvolutions of probability density functions and, as noted in the source, would imply the bound $\Delta(\varepsilon)\geq 0.651\varepsilon^2$.