  Recently, Gilmer proved the first constant lower bound for the union-closed
sets conjecture via an information-theoretic argument. The heart of the
argument is an entropic inequality involving the OR function of two i.i.d.\
binary vectors, and the best constant obtainable through the i.i.d.\ coupling
is $\frac{3-\sqrt{5}}{2}\approx0.38197$. Sawin demonstrated that the bound can
be strictly improved by considering a convex combination of the i.i.d.\
coupling and the max-entropy coupling, and the best constant obtainable through
this approach is around 0.38234, as evaluated by Yu and Cambie. In this work we
show analytically that the bound can be further strictly improved by
considering another class of coupling under which the two binary sequences are
i.i.d.\ conditioned on an auxiliary random variable. We also provide a new
class of bounds in terms of finite-dimensional optimization. For a basic
instance from this class, analysis assisted with numerically solved
9-dimensional optimization suggests that the optimizer assumes a certain
structure. Under numerically verified hypotheses, the lower bound for the
union-closed sets conjecture can be improved to approximately 0.38271, a number
that can be defined as the solution to an analytic equation.
