Whether the statement is true or not does not matter for purposes of the fair report privilege: even if the witness whose testimony you relied on is later convicted of perjury, the privilege still applies if you accurately reported and attributed the testimony he provided in the first place. It would apply even if you had knowledge that the witness was lying in his testimony. The purpose of the privilege is to protect statements or facts from public sources that are newsworthy in and of themselves, regardless of their veracity.
Generally, courts will follow rules of accuracy that echo the "gist" and "sting" rule developed to test for "substantial truth." See the section on Substantial Truth for more information.
But what is critical is that you accurately report (or abridge fairly) the information: reporting that the witness said the defendant deliberately burned down the house when the witness had only said that the defendant accidentally dropped a match would not be protected by the fair report privilege. Be particularly careful when you are "translating" complex legalese. Further, be careful not to use quotations selectively. For example, if a witness in her testimony said she saw the defendant rob the store, then corrects herself thirty minutes later in the same testimony to indicate that she had really not seen the robbery, quoting only the first part would likely fall outside the fair report privilege.
In general, courts will look at whether you acted in "good faith," looking far more favorably at an honest mistake that was made in condensing a long, complex statement or document than at selective quotation that may be perceived as maliciously intending to portray the subject in the least favorable light possible. Not every fact must be included, but many courts will find the privilege lost if the overall reporting is too one-sided.