Recharacterization of ACP test failures to after-tax in a 403b plan?

Is it possible to take an HCE's ACP testing refund and have it recharacterized to an after-tax "bucket" within a 403b plan?  I would imagine the plan would need to issue a 1099R reporting the taxable event for the required distribution amount.  

Thanks so much for your assistance!

Does the plan allow for after tax contributions (NOT Roth)?

Not currently, but considering adding this provision to accomplish the above, if possible.  I understand that any after-tax contributions have to be considered in the ACP test.  Since this client usually fails the ACP test, I'm wondering if this would be helpful at all.

I am not sure if there's a way to "recharacterize" the ACP refund to be an after-tax contribution or if it would have to be first distributed to the HCE and then have the HCE redeposit it into the plan as an after-tax contribution.  Whether it can be "recharacterized" or if it has to be a 2 step process, would the after-tax contribution be considered in the same testing year as the refund occurred or the year in which it is contributed?

Thank you!