The $3n+1$ function is given by $T(n)=n/2$ for $n$ even,
$T(n)=(3n+1)/2$ for $n$ odd.
Given a positive integer $a$, another number $b$ is a called a predecessor of $a$ if some iterate $T^\nu(b)$ equals $a$.
Here some ideas are described which may lead to a proof
the set of predecessors of $a$ has positive lower asymptotic density,
for any positive integer $a\ne 0 $ mod 3.
Three unbridged gaps in the argument are formulated as conjectures.