You need to manually make the job size longer in the Y direction (to the top of the wrap vector) and make the Y dimension of the component the full height of the updated job size. This will result in the tapered component "wrapping" more than 360 degrees around the cylinder.
This change in job size isn't required when putting a spiral on a simple cylinder, but since you want to project the spiral onto a taper, you need to make sure there's a taper underneath the spiral for the full distance of the cut (not sure that sentence made sense, but hopefully you get the gist).
See the example pictures below, from your original file.
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