Hereditary normality of a space of the form $\mathscr F(X)$

Assuming the continuum hypothesis we construct an example of a nonmetrizable compact set $X$ with the following properties
1) $X^n$ is hereditarily separable for all $n\in\mathbb N$,
2) $X^n\setminus\Delta_n$ is perfectly normal for every $n\in\mathbb N$, where $\Delta_n$ is the generalized diagonal of $X^n$, i.e., the set of points with at least two equal coordinates,
3) for every seminormal functor $\mathscr F$ that preserves weights and the points of bijectivity the space $\mathscr F_k(X)$ is hereditarily normal, where $k$ is the second smallest element of the power spectrum of the functor $\mathscr F$; in particular, $X^2$ and $\lambda_3X$ are hereditarily normal.
Our example of a space of this type strengthens the well-known example by Gruenhage of a nonmetrizable compact set whose square is hereditarily normal and hereditarily separable.