No submaximal topology on a countable set is $\mathbf{T}_{\mathbf{1}}$-complementary

Two $T_{1}$-topologies $\tau$ and $\mu$ given on the same set $X$, are called transversal if their union generates the discrete topology on $X$. The topologies $\tau$ and $\mu$ are $T_{1}$-complementary if they are transversal and their intersection is the cofinite topology on $X$. We establish that for any connected Tychonoff topology there exists a connected Tychonoff transversal one. Another result is that no $T_{1}$-complementary topology exists for the maximal topology constructed by van Douwen on the rational numbers. This gives a negative answer to Problem 162 from Open Problems in Topology (1990).