In order for "round earth" conspiracy to be true every self respecting scientist from the first century to today would've had to have been part of the conspiracy.
From wikipedia:
By the time of Pliny the Elder in the 1st century, however, the Earth's spherical shape was generally acknowledged among the learned in the western world. Ptolemy derived his maps from a curved globe and developed the system of latitude, longitude, and climes. His writings remained the basis of European astronomy throughout the Middle Ages, although Late Antiquity and the Early Middle Ages (ca. 3rd to 7th centuries) saw occasional arguments in favor of a flat Earth.
The modern misconception that people of the Middle Ages believed that the Earth was flat first entered the popular imagination in the nineteenth century, thanks largely to the publication of Washington Irving's fantasy The Life and Voyages of Christopher Columbus in 1828.