In this question Proving a sequence of integrals increases (iterated minimax distributions)

Pietro Majer proved that

$$\int_0^1F_n(x) dx \leq \int_0^1F_{n+1}(x) dx$$ when $$F_n(x) = (1-(1-F_{n-1}(x))^c)^c$$ and $F_0(x) = x$ and $n$ and $c$ are integers, $n\geq 1$ and $c \geq 2$$

In other words, that the sequence $\int_0^1F_n(x)dx$ increases with $n$.

I would really like to also be able to show that the sequence increases at a strictly decreasing rate: i.e. that $$\int_0^1F_{n+2}(x) dx - \int_0^1F_{n+1}(x) dx < \int_0^1F_{n+1}(x) dx - \int_0^1F_{n}(x) dx $$

Can anyone help me out? I wasn't sure if it was more appropriate to add comments in the original question, or to start a new question. Hopefully I made the right call. Thank you!