For your integrity's sake, you need to pay for the 78 days notice period. I don't think your HODs agreement with you on your notice period serves to replace the statutory requirement from the company. Remember, the notice period is supposed to be a period when the company readjusts, and possibly find a replacement for the resigning staff. Although i feel 90 days is unrealistic, it really does take time, energy and money to find replacements. This is why you are made to pay for that inconvenience.
As someone suggested, you could have used your leave days as notice period.
As someone suggested, you could have used your leave days as notice period.