Using $(3p+1)/2$ starting with $p = 44102911$, we find an ordered set of $8$ primes. By computer, we find that this is the only ordered set of $8$ primes $< 300000000$ primes.
The primes:
$\{44102911, 66154367, 99231551, 148847327, 223270991, 334906487, 502359731, 753539597\}$
When we take the differences and remove the common factor, $344554$, we have:
$\{64, 96, 144, 216, 324, 486, 729\}$
As you can see, this is the $7th$ row of Nicomachus's Triangle. A036561

Question: What would it take to show that a count of $8$ is the upper bounds for this ordered set?
Terry Tao has a recent blog about this.
Collatz on arxiv and another Collatz on arxiv.
Set of $7$ primes
$\{89599, 134399, 201599, 302399, 453599, 680399, 1020599\}$
When we take the differences and remove the common factor, $1400$, we have:
$\{32, 48, 72, 108, 162, 243\}$
As you can see, this is the $6th$ row of the triangle.