I'm sorry for not answering your question directly, but would like to raise an alarm regarding the proposed design.
Consider someone is after an item as such:
Menu7 > Secondary Menu4 > Sub4
On a desktop (based on your screen shot) it will take them 3 clicks.
On a mobile device, it will take them:
- 1: Move to Menu 3+4
- 2: Move to menu 5+6
- 3: Move to menu 7+8
- 4: Click on 7
- 5: Click on Hamburger Icon
- 6: Click on Secondary Menu4
- 7: Click on Sub4.
Since the display of a mobile device is small, mobile devices tend to involve more interaction gestures. Thus, the popular recommendation is to try and reduce the number of interactions as much as possible (the number will be high regardless).
I don't think 7 clicks is to be considered usable.
I do understand that a similar interface on both desktop and mobile yields higher consistency, but while the debate is ongoing, many will prioritise usability over consistency. Further, research has shown that designers are much more sensitive to consistency issue compared to real users.
One key concept in interaction design is to try and provide the required content, yet allowing users to perform their tasks with the least amount of (interaction) effort.
The problem with the scrolling menu is that it increases interaction effort, and it is not ideal for learnability - users don't see what other options are there without having to put effort into finding out.
Given we take it for granted users typically want to get on with their task, such a behaviour is not ideal.
If it is a top priority to keep the various interfaces consistent, I would suggest just providing an hamburger icon on both - and upon click big (or the whole) navigation will be revealed. But as in design, proper solutions come from weighted diversities - not a single proposal.