I just played with this CodePen and it seems that if you set the background-color to transparent for all three properties below (and in this example also remove box-shadows), the scrollbar is not visible at all:

 

#style-1::-webkit-scrollbar-track {
  //    -webkit-box-shadow: inset 0 0 6px rgba(0,0,0,.1);
  background-color: transparent;
}

#style-1::-webkit-scrollbar {
  background-color: transparent;
}

#style-1::-webkit-scrollbar-thumb {
  //    -webkit-box-shadow: inset 0 0 6px rgba(0,0,0,.3);
  background-color: transparent;
}

 

This was tested in Chrome Desktop, Chrome for Android, and iOS Safari (v8.4) on an iPhone 6+.

I would however recommend for user experience (usability/accessibility) to keep the scrollbar visible though, as even I, knowing what I was doing, got a bit confused when there was no scrollbar.

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