Browser cookie blockers prevent public reports from showing up in iframes

We would like the respondents to be redirected to the public report at the end of a survey. This process is working as expected in any browser, including in incognito mode. However, when I put the survey in an iframe in a different website, the survey shows up, but the redirecting step to the public report fails. After some poking around, the problem seems to be coming from the third-party cookie blockers in the browsers. Ideally we wouldn't want the users to have to open the report in a different tab, or give them instructions about allowing 3rd party cookies. Is there a work around? Am I missing something obvious? I've tried Chrome, Safari and Brave -- all them have the same behavior.
Many thanks in advance for your help!