Missing access token under ./auth/me using Easy Auth

Question

I'm creating an Azure Static Web App with the built-in (Easy Auth) authorization to the Azure AD. This works well, but I would also like to authorize access to user-specific data in another API (AWS, but I have registered also this as an app in Azure AD). My thought was to do this with an access token, but heading to the /.auth/me only gives me the following information:

identityProvider
userId
userDetails
userRoles: [ "authenticated", "anonymous" ]
claims: aud, iss, iat, nbf, exp, aio, c_hash, name, nonce, .../objectidentifier, preferred_username, rh, .../nameidentifier, .../tenantid, uti, ver

The available cookies are: AppServiceAuthSession. AppServiceAuthSession1 and StaticWebAppsAuthCookie

I have set "accessTokenAcceptedVersion": 2 in the Azure AD app, as well as enabling both access and id tokens.

My staticwebbapp.config.json has this in it (with <TENANT_ID> and <API_AD_APP> replaced) :

    "identityProviders": {
      "azureActiveDirectory": {
        "registration": {
          "openIdIssuer": "https://login.microsoftonline.com/<TENANT_ID>/v2.0",
          "clientIdSettingName": "AZURE_CLIENT_ID",
          "clientSecretSettingName": "AZURE_CLIENT_SECRET"
        },
        "login": {
          "loginParameters": [
            "scope=openid profile email offline_access api://<API_AD_APP>/user_impersonation",
            "response_type=code id_token"
          ]
        }
      }
    }
  },

In the API app AD I have set added the user_impersonation scope under "Expose an API", and authorized my main AD app to use it.

Is it possible to get hold of the access cookie this way, or do I have to switch to requesting it using MSAL instead?

EDIT:

There is now an issue about enabling this functionality:
https://github.com/Azure/static-web-apps/issues/794