It was previously shown that $$H\ge cG,\tag{1}$$ where $c:=1/14334$, $$G:=E|X-Y|,\quad H:=E|X-Y|-\tfrac12\,E|X+Y-2Z|,$$ and $X,Y,Z$ are independent random variables with the same log-concave density.
It is clear that the constant factor $c=1/14334$ is not optimal.
Question: What is the best (that is, the largest) possible constant factor $c$ in (1)?
The following might turn out to be of help:
Conjecture: The best possible value of $c$ is $1/9$, attained when $X$ has an exponential distribution (which would thus represent the least favorable case).
