In this article (page 2) , the authors say:

"it is expected, based on known examples, that Calabi–Yau threefolds of large Picard rank are always elliptically fibered, perhaps after flopping a finite number of curves."

They did not provide any references for this claim in the paper. Any supporting examples or references for this claim?

K3 surfaces of large Picard rank ($\ge 5$) are always elliptically fibered but three dimensional situation is quite different.