On how many Fibonacci words is the Plancherel measure concentrated

On the set ${\mathbb{Y}\mathbb{F}}_n$ of words in $1$ and $2$ with digit sum $n$, we consider the probability distribution ${\mu }_P$, corresponding to the Plancherel measure on paths in the Young–Fibonacci graph. We study the asymptotics as $n$ grows of the entropy of the distribution ${\mu }_P$ and of the minimum number of words on which ${\mu }_P$ is almost entirely concentrated.