Read the abstract of the actual study. The abstract results summarize that the telomere length issue was not correlated with non-carbonated sugar-sweetened beverages, or carbonated non-sugar-sweetened beverages. It's only when both traits are present that this happens.
Sadly there are no more detailed results to look at HFCS or Sucrose. My guess is the study used HFCS pops, but bear in mind that numerous juice brands use HFCS as a sweetening additive, which would count here as a non-carbonated sugar-sweetened beverage. The next question is what's the interaction that causes this from these two traits?