For all purchase orders which are placed through COIN, the units which are allotted will always be in demat mode. So, you can check your demat account statement which is a valid proof of investment. The AMC provides the folio details for all purchase transactions and the same is reflected in your portfolio on COIN.
But, the problem with most AMCs is that they are not able to provide you a statement or validate your investments if the units are held in demat mode so you would have faced an issue mentioned in your post.
Hope this day never comes 
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But, in such a scenario you can approach the depository to help you out with redeeming the units since the units are in demat account. In this case, the depository would be CDSL.