I believe that the links between sugar and acne hinge on two main points: insulin and allergies, both linked to inflammation.
Obviously, consumption of sugars (refined or not) will lead to a spike in blood sugar and a release of insulin. This sudden release of insulin causes an inflammatory response by your immune system; inflammation is what causes your blocked pores to break out in the nasty red pimples or whiteheads.
Allergies are more common when consuming foods that contain sugar, such as most refined sugar products (often containing gluten) or even fruit. Obviously gluten sensitivity can cause an inflammatory outbreak of acne, but fruit can do the same if the user suffers from fructose intolerance or fructose malabsorption. Fructose can cause an inflammatory response that leads to skin rashes and acne in certain portions of the population (#s vary widely from 5% to supposedly up to 30%; there are tests for it, but you can usually figure it out just by doing an elimination diet)
If you're dead set on consuming sugar and still want to minimize your chance of acne, you might want to look into some other factors for acne:
-Make sure you've checked out dairy to see if it's a factor in your acne. A lot of people find that cutting out dairy (or even just milk) can have a huge effect on their acne.
-Ensure you're getting enough Omega 3 Fatty Acids. O3 fatty acids are integral to skin health and a lack of them can lead to breakouts.
-If it's winter time and/or you spend a lot of time indoors, be prepared to use some sort of moisturizer. I'm certain that back in the day Grok didn't use moisturizer, but he also didn't have to deal with the drying effect of an air conditioner blasting all day.