The answer is definitely yes for so-called "One-Time Codes." But I have the same question as you for the more flexible "Custom Codes."

The help page on Promo Codes says clearly that"Custom codes can be redeemed only from within your app, while one-time codes can be redeemed through both your app and the Play store."

But later in the same doc, it says—without differentiating between one-time and custom codes—simply that "You can also share a promo code by generating a URL that sends the user to the Google Play Store and auto-populates the Enter code field."

So does anyone know whether deep linking works for custom codes? (I guess I hope they do but suspect they don't.)