First, some notation. Define $T(n)$ over $n\in \mathbb{N}$ as: $$ T(n) = \left\{ \begin{array}{} 3n+1, & \text{if $n$ is odd}\ \\ n/2, & \text{if $n$ is even} \end{array} \right. $$
And let: $$ \pi_a(x) = \#\{ n : n\leq x \mbox{ and } T^j(n) = a \mbox{ for some $j$ }\} $$
Of course, the classic Collatz conjecture is that $\pi_1(x) = x$ for all $x$.
A succession of papers (see references below) have established bounds on $\pi_a(x)$ of the form: For $a \not\equiv 0 \; (\mbox{mod } 3)$, $$ \pi_a(x) \geq x^\gamma, \; \forall x \geq x_0(a) $$ The cases where $a \equiv 0\; (\mbox{mod } 3)$ are generally uninteresting for these questions since we know that the set $\{n : T^j(n) = a\}$ is precisely $\{2^k a : k=0,1,2,\ldots\}$.
In [1], the authors state the following conjecture (though I'm uncertain if they originated this conjecture or if there are earlier citations for it):
$$ \text{Conjecture A}:\mbox{For each $a\not\equiv 0 \;(\mbox{mod } 3)$, there is a positive constant $c_a$ such that}\\ \pi_a(x) \geq c_a x, \; \forall x \geq a $$
There appears to be at least a tiny bit of empirical data to support this conjecture. Some arithmetic reveals that the last odd number (not including 1) in a Collatz trajectory must be of the form ${2^{2k}-1}\over{3}$. I took a pseudo-random sample of 10,000,000 numbers in the interval $[0, 2^m-1]$ for each $m \in \{64,128,256,512,1024,2048\}$ and recorded the last odd number in each trajectory. The results are summarized in the table below: $$ \begin{array}{r|c|c|c|c|c|c} \; & m=64 & 128 & 256 & 512 & 1024 & 2048 \\ \hline k=2 & 9377219 & 9377605 & 9376867 & 9377728 & 9378367 & 9377611 \\ 4 & 238003 & 237079 & 238246 & 237240 & 237296 & 237829 \\ 5 & 378816 & 379476 & 379021 & 379013 & 378382 & 378506 \\ 7 & 787 & 782 & 783 & 760 & 808 & 831 \\ 8 & 4839 & 4736 & 4762 & 4916 & 4814 & 4896 \\ 10 & 311 & 295 & 310 & 322 & 317 & 307 \\ 11 & 21 & 24 & 9 & 19 & 15 & 17 \\ 13 & 3 & 2 & 2 & 1 & 0 & 2 \\ 14 & 1 & 1 & 0 & 1 & 1 & 1 \\ \end{array} $$
Question: Is there any intution or heuristic that supports Conjecture A? Or, if not for the conjecture in full generality, at least for these set of "last" odd numbers of the form ${2^{2k}-1}\over {3}$, $ k\not\equiv0(\mbox{mod }3)$?
By "intuition or heuristic", I'm thinking of perhaps something in the same vein as this probabilistic heuristic for why the Collatz conjecture itself should be true.
Applegate, David; Lagarias, Jeffrey C., Density bounds for the (3x+1) problem. I: Tree-search method. II: Krasikov inequalities, Math. Comput. 64, No. 209, 411-426, 427-438 (1995). ZBL0820.11006.
Crandall, R. E., On the ”3x+1” problem, Math. Comput. 32, 1281-1292 (1978). ZBL0395.10013.
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Krasikov, Ilia; Lagarias, Jeffrey C., Bounds for the (3x+1) problem using difference inequalities, Acta Arith. 109, No. 3, 237-258 (2003). ZBL1069.11011.
Sander, J. W., On the ((3N+1))-conjecture, Acta Arith. 55, No. 3, 241-248 (1990). ZBL0707.11017.
Wirsching, Günther, An improved estimate concerning (3n+1) predecessor sets, Acta Arith. 63, No. 3, 205-210 (1993). ZBL0804.11022.
