ABG is an Ahmedabad-based shipbuilding company, while Mandhana is a multi divisional textile firm.
The latter's subsidiary Mandhana Retail Ventures manufactures, retails and distributes Bollywood actor Salman Khan's Being Human brand.
In a circular, BSE said these firms will "not be suspended with effect from March 27, 2017 as the companies have complied with the Regulation 33 of Sebi (Listing Obligations and Disclosure Requirements) Regulations, 2015".
The Regulation 33 pertains to submission of financial results.
In case the firms comply with norms by March 21, they would not face the suspension, the exchange had said.
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