  We introduce a forcing that adds a $\square(\aleph_2,\aleph_0)$-sequence with
countable conditions under CH. Assuming the consistency of a weakly compact
cardinal, we can find a forcing extension by our new poset in which both
$\square(\aleph_2,<\!\aleph_0)$ and $\square_{\aleph_1,\aleph_0}$ fail in the
forcing extension.
