Yes, they were related, respected one another, and studied together. But the Steipler, it seems to me, did not accept the Chazon Ish as his general Rebbe. There is a well-known letter by the Chazon Ish addressed to the Steipler Illuy, chastising him for using an oath unnecessarily in writing. The Rishonim must have had specific reasons for doing so, occasionally.*And “Sahadei Bemeromim” is an oath.
And yet, “Iggros Ureshimos Kehillos Ya’akov” (Part 1, page 71) records another letter containing “Sahadei Bemeromim”, dated year 5732.
*Note: The Chazon Ish did not say “The Rishonim could, but we cannot“, as Rabbi Elyashiv “explained” his decision to ban books purportedly repeating the views of certain Rishonim on matters of faith, see here.