My question regards properties of the **f-bar** metric $\bar{f}$ defined for shift invariant measures on $\mathscr{A}^\infty$
where $\mathscr{A}$ is a finite alphabet. The definition of the $\bar{f}$ metric mimics the definition of the **d-bar** metric $\bar{d}$ and shares many properties with the later. Unfortunately the results are scattered and the sources use different variants of the definition. I would like to know whether the following result holds true (the analogous result for $\bar{d}$ is known).

Given two $n$ words $u=u(0)u(1)\ldots u(n-1)$ and $w=w(0)w(1)\ldots w(n-1)$ over $\mathscr{A}$ we define $$ \bar{f}_n(u,w)=1-\frac{k}{n}, $$ where $k$ is the largest integer $\ell$ such that for some $0\le i_1<i_2<\ldots<i_\ell<n$ and $0\le j_1<j_2<\ldots<j_\ell<n$ we have $u(i_s)=w(j_s)$ for $s=1,\ldots,k$. For two infinite sequences $x=x_0x_1x_2\ldots$ and $y=y_0y_1y_2\ldots$ over $\mathscr{A}$ we set $$ \bar{f}(x,y)=\limsup_{n\to\infty} \bar{f}_n(x_0x_1\ldots x_{n-1},y_0y_1\ldots y_{n-1}) $$

Let $\mu$ and $\nu$ be ergodic shift invariant measures on $\mathcal{A}^\infty$. By $\mu_n$, respectively $\nu_n$ we denote the restriction of $\mu$, respectively $\nu$ to the set of all $n$-cylinders,that is, the measures that $\mu$ and $\nu$ respectively define on $\mathcal{A}^n$ via the projections onto first $n$ coordinates. Let $J_n(\mu,\nu)$ denote the set of all measures $\lambda_n$ on $\mathcal{A}^n\times \mathcal{A}^n$ whose marginals are $\mu_n$ and $\nu_n$.

Define
$$
\bar{f}_n(\mu,\nu)=\inf_{\lambda_n\in J_n(\mu,\nu)}\int_{\mathcal{A}^n\times \mathcal{A}^n}\bar{f}_n(u,v) \lambda_n(u,v).
$$
The **f-bar** distance between measures is given by
$$
\bar{f}(\mu,\nu)=\sup_{n\ge 1} \bar{f}_n(\mu,\nu)=\lim_{n\to\infty}\bar{f}_n(\mu,\nu).
$$

And here is the question: Assume that $x$ is a generic point (typical sequence) for $\mu$ and $y$ is a generic point (typical sequence) for $\nu$. Is it true that $$ \bar{f}(\mu,\nu)\le \bar{f}(x,y)? $$