everyone agrees with 539 BCE for the fall of Babylon, and 537 BCE as the date of the return from exile.
Who is everyone? 537 BCE is not in the Bible... The specific amount of time between Cyrus conquering and the Jews going back is not mentioned, since the account merely mentions which month they all went back... the WT says it plainly when they say "it is reasonable to believe that the Israelites went back in 537 BCE...
*** w64 2/1 p. 80 pars. 5-6 The Comely Feet of the Messengers ***
Jehovah God had decreed that their land should rest without inhabitants for seventy years. Its seventieth year of sabbath-keeping began in the early autumn of 538 B.C., about the time of the Jewish Day of Atonement. Cyrus’ decree of liberation was published in the first year of his reign. Babylon fell before him in 539 B.C., and Bible chronologers figure that his first regnal year began in the spring of the year following, on Nisan 1, 538 B.C., which would be six months before the seventieth year of sabbath-keeping of the land of Judah began. If Cyrus had issued his decree early in 538 B.C. as his first regnal year, the exiled Jews could have made the journey back to their land before its seventieth year of sabbath-keeping began, which would have been contrary to Jehovah’s prophecy. Or they might have had to journey through the rainy winter months of the year, which would be from October through March. This would not be too good.
6 By the calculation above, Cyrus’ first regnal year would end on March 5 of 537 B.C., or toward the end of the rainy season. Doubtless he considerately issued his decree near the end of the rainy season, shortly before March 5 of 537 B.C., and in that way the exiled Israelites could make arrangements to travel in the convenient dry season from April through September. They evidently made the four-month-long journey from Babylon during the dry months, for by October they were back and settled in their beloved homeland, before the first day of their seventh lunar month.—Ezra 3:1.
But it could have been 538 or 536 for all we know... can you count the number of possibel statements above (they are highlighted for your convenience...)
The proof is then: 537 must be true, since if it is not, then Jehovah lied to us about 607... circular reasoning at its best!
Another point: What they fail to say is: "Evreyone except us agree with 587 too, we are the only idiots that don't agree with the facts!"
A@G