SOLVED - Thank you to hints from Jared Smith and Pointy.

async function getKeyIndex(storeName) {
    return new Promise((resolve, reject) => {
        const dbStore = [];
        let c = 0;
        const objectStore = db.transaction(storeName).objectStore(storeName);
        const cursorRequest = objectStore.openCursor();

        cursorRequest.onsuccess = (event) => {
            const cursor = event.target.result;
            if (cursor) {
                console.log("Name: " + cursor.key + " - id: " + cursor.value.id);
                dbStore.push(cursor.value.id);
                c++;
                cursor.continue();
            } else {
                resolve(dbStore);
            }
        };
    });
}

Fired by:

        keyIndex = await getKeyIndex('conversations');
        console.log("Database keys " + keyIndex);