Sum-Tower of Hanoi

Below is a simplified diagram of the Tower of Hanoi with 4 rods and a stack of $n$ disks in ascending order of weight:

Hanoi Jr. is interested in determining the maximum value of $n$ such that all the $n$ disks can be transferred from "Start" to "Finish" following all the rules including an additional rule explained below.

In addition to the standard rules of Tower of Hanoi where no disk may be placed on top of a smaller disk, for the two red-colored rods in the middle, no disk may have more sum of weights placed on it than its own weight. That is, if the stacked weights on either rod at any step are $a_1< a_2< \cdots< a_m$ with $m\le n,$ then for any $1 \leq k \leq m-1,$ it must hold that $\sum\limits_{i=1}^k a_{i} \leq a_{i + 1}.$

What is the maximum possible value of $n?$