Monotonicity of Odd Powers under a Tracial State

In the setting of Noncommutative Laws, Couplings and the Wasserstein Distance: Standing Notation, let $m\in\mathbb{N}$, let $\lambda$ be a tracial state on $\mathcal{P}_{m}$ and let $a,b\in\mathcal{P}_{m,\mathrm{sa}}$. For $c\in\mathcal{P}_{m}$ and $n\in\mathbb{N}$, $c^{n}$ is the product of $n$ factors $c$, that is, the product along the word of length $n$ all of whose letters are $1$, for the $1$-tuple $(c)$.

¶For every $k\in\mathbb{N}$, with $n=2k-1\in\mathbb{N}$, the number $\lambda\bigl((a^{n}-b^{n})(a-b)\bigr)$ is real and nonnegative.