Log-concave probability measure with slowest decay

Question

Let $X$ be a real valued random variable with log-concave distribution $\mu$. For each $x \in {\mathbb R}$, let $$ \phi_\mu(x)=\min\limits_{c\in{\mathbb R}}E[e^{c(X-x)}] $$ be the minimal value of the moment generating function of a shifted random variable $X-x$. Let $$ b_\mu = E[\phi_\mu(X)]. $$ The constant $b_\mu$ belongs to the interval $(0,1)$, is invariant under affine transformations of $X$, and in some sense measures the "rate of decay" of $X$ (the higher $b_\mu$ the slower decay). The question is: which log-concave distribution $\mu$ maximizes $b_\mu$? Is it Laplace distribution? As a partial progress, it would be interesting to find any upper bound $b<1$ such that $b_\mu \leq b$ for all log-concave $\mu$.