I'm not sure where the "three times" came from. Maybe a typo/misunderstanding of the following statement from the source article?
"[...] ultra-large-capacity optical fibers will be needed in the future, where the data traffic demand is expected to increase by 3 orders of magnitude (x1,000 times)"
I believe it's 3X the throughput of other designs using the same cable. Those cables (38 core) are not yet deployed, so it's 1,000 times the throughput of currently deployed systems. From the paper:
"The measured transmission capacity for each core ranged from ~0.3 to 0.7 petabits per second leading to a total transmission capacity of 22.9 petabits per second. The achieved data-rate includes an overhead for an implemented forward-error correction code with the demonstration showing up to 24.7 Pb/s can be achieved with better optimized coding. This is more than 1,000 times the data-rate of currently deployed optical fiber communication systems."
That's right before the next paragraph that you quoted in part:
"While uncoupled four-core MCF is suitable for early adaptation, further improvement of the telecommunication infrastructure using ultra-large-capacity optical fibers will be needed in the future, where the data traffic demand is expected to increase by 3 orders of magnitude (x1,000 times). This study demonstrates the first successful combination of multi-band WDM and SDM employing a multicore multimode fiber, which is key to the realization of future ultra-large-capacity optical fiber communication networks."