  In sequential hypothesis testing, Generalized Binary Search (GBS) greedily
chooses the test with the highest information gain at each step. It is known
that GBS obtains the gold standard query cost of $O(\log n)$ for problems
satisfying the $k$-neighborly condition, which requires any two tests to be
connected by a sequence of tests where neighboring tests disagree on at most
$k$ hypotheses. In this paper, we introduce a weaker condition,
split-neighborly, which requires that for the set of hypotheses two neighbors
disagree on, any subset is splittable by some test. For four problems that are
not $k$-neighborly for any constant $k$, we prove that they are
split-neighborly, which allows us to obtain the optimal $O(\log n)$ worst-case
query cost.
