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 are always elliptically fibered but three dimensional situation is quite different.