Basically you need to...

- Remove that guassian blur shadow. That will never be vector (no "blur" will)

- Expand the strokes so they ae shapes and then combine them. (You'll need to remove the gradients from the strokes in order for them to expand correctly)

- Once the strokes are combined, simply add a gradient fill across the shape and a black stroke to mimic the black outline you originally had.

- Add a secondary fill or a duplicate offset shape to create the black shadow, but don't apply any blur effects to it.

The image can easily be created so it remains flat vectors and does not rasterize, but you need to simply rethink construction of the objets entirely.

(Gradient may not match original gradients exactly, I was merely getting close)

My image above is one gradient fill across all the shapes. Your original image had a separate gradient stroke on each and every shape with the gradients, painstakingly, matched at the edges. In many cases you ahd 2 or 3 strokes overlapping each other for the same shape.

Why can't you simply use one gradient fill as I've done above? You don't need strokes.

As long as you aren't adding any gradients on the strokes and not adding any bluring or drop shadow effect, it should remain vector.

You can use Window > Flattener Preview.  Choose All Effected Objects from the drop down and click Refresh... if you see red shapes.. those are problem areas. No red highlights... you're good to go. Try this on your original file and you'll see what it indicates.

If you can't select anything from the Flattener Preview drop down menu, it means there's nothing to flatten.  And if nothing is flattened, the art will remain vector exactly as you see it when saved to an eps format.