If you file a dispute with one CRA, no, the CRA is not charged with the responsibility of notifying the other CRAs
However, that does not mean you must file the same dispute with each CRA. If the furnisher's investigation results in correction or deletion, they have the burden of reporting the same result to all CRAs.
If they fail to report the result to all CRAs, call and remind them of their obligation. If informal attempts fail, you can then file a direct dispute with them, with the inaccurcay being failure to have complied with section 623(b)(D).
"FCRA 623(b) Duties of Furnishers of Information upon Notice of Dispute
(1) In general. After receiving notice pursuant to section 611(a) of a dispute with regard to the completeness or accuracy of any information provided by a person to a consumer reporting agency, the person shall
(A) conduct an investigation with respect to the disputed information;
(B) review al relevant information provided by the consumer reporting agency pursuant to section 611(a)(2):
(C) report the results of the investigation to the consumer reporting agency;
(D) if the investigation finds that the information is incomplete or inaccurate, report those results to all other consumer reporting agencies to which the person furnished the information and that complile files on consumers on a nationwide basis;..."
