calculus of variations - Maximizing the volume in a family of subsets of a cube

Starting from a question in probability, one is eventually lead to the following optimization problem.

Let $I:=[0,\, 1],$ and let $A$ be a Lebesgue measurable subset of the $n$-dimensional cube, $Asubset I^n.$ Consider, correspondingly, the set
$$hat A:= {xin I^{n+1}:\, (x_1,dots,x_n)in A,\, (x_2,dots,x_{n+1})notin A}=Atimes I\,cap\, I times A^c.$$

Problem. Maximize the $(n+1)$-dimensional Lebesgue measure
of $hat A$ over all measurable
$Asubset I^n$:
$$lambda_n:=sup_{Asubset I^n}verthat Avert.$$

If $n=1,$ we have $|hat A|=|A|(1-|A|),$ whence $lambda_1=1/4.$ For $n=2$ the maximizing set is the triangle below the diagonal, giving $lambda_2=1/3.$ The sequence $lambda_n$ is increasing, and converges to $1/2.$ If $n$ is even, one finds $$lambda_n=frac{1}{2}left(1-frac{1}{n+1}right).$$
(I will edit and provide the details of the computation at request). However, as a consequence of a computation by Trotter and Winkler (Ramsey theory and sequences of random variables, Probability, Combinatorics and Computing 7 (1998), 221-238), the formula can't hold true for all odd $n,$ for one has $lambda_5>frac{1}{2}left(1-frac{1}{6}right)=5/12.$

I would be very grateful for any suggestion or reference useful to shed light on the case of odd $n.$