I have looked at all the observations recorded on VAT4956 and applied them to both 588/587 BCE and 568/567 BCE using the same astro program the Watchtower's researchers used. Following rigorous criteria the results are zero matches for 588/587 BCE as Neb II 37th year. But I found 34 matches out of 36 for 568/567 BCE.
The only way to get few lunar position matches for 588/587 BCE (using the WT scheme) is by ignoring the fact that you are using a fabricated calendar that is not attested by the thousands of known dated tablets, creating thus a calendar year that starts at the "wrong month" (a month late). You can get few more lunar matches by ignoring the non-visibility of the first crescent for the first day of the month on the fabricated calendar, creating thus a calendar year with at least three months that start "at the wrong day of the month" (one day too soon). Even overlooking all the trickery of the WT, the best you can do is get no more than 8 matches for 588/587 BCE.
Regarding time intervals and planetary positions you have a 100 % success rate for 568/567 bce and 100 % failure rate for 588/587 bce.
CONCLUSION: 587/586 BCE is the date for the fall of Jerusalem according to the Bible itself.