Functional strong slicing conjecture

Let $L_f$, $\widetilde{L}_f$, and $\widehat{L}_f$ be the three functional isotropic constants defined for integrable centered log-concave functions. Define

$$f_0(x)=e^{-\sum_{i=1}^n x_i}\chi_{[-1,+\infty)^n}(x),\quad f_1(x)=e^{-\sum_{i=1}^n|x_i|},\quad f_\infty(x)=\chi_{[-1,1]^n}(x).$$

Functional strong slicing conjecture. For every integrable log-concave $f:\mathbb{R}^n\to\mathbb{R}_+$,

$$L_f\leq L_{f_0}=1,\qquad \widetilde L_f\leq\widetilde L_{f_1}=\frac1{\sqrt2},\qquad \widehat L_f\leq\widehat L_{f_0}=e^{-1}.$$

The three assertions hold in dimension one, while the supplied text presents their higher-dimensional validity as conjectural; the third assertion implies the first via the displayed comparison inequalities.