It is possible that the star jasmine have "armored scale" aka "hard-bodied scale", a sucking insect pest similar to the soft bodied scale that you may be more familiar with.
Control is possible, but timing of spraying is critical for effective control and using an inappropriate material or wrong timing will be a waste.
The immature scale insects are called "crawlers" and they are vulnerable to contact sprays like insecticidal soap or chemical insecticide, and smothering oils like neem oil or horticulural oil. But the adult scales are settled in and feeding and are covered with a protective coating, so the only access for control is with "systemic" insecticide.
You can keep up with 70% neem oil repeated spraying and eventually you will cover new hatches of crawlers and the infestation will come under control, especially after the primary crawler periods of spring and late summer-fall. But for more immediate control you will need to use specific systemic chemical control for armored scale. Some systemics are effective for other pests like aphids, but not as effective for armored scale.
An effective product used by professional applicators is 'Safari' insecticide, applied as a water soluble soil drench for root uptake. The active ingredient is dinotefuran. Its kind of expensive but doesn't take much and is quick acting, highly effective with a one time application. Follow up with preventive neem oil apps through the season next year.