On non-K\"ahler compact complex manifolds with balanced and astheno-K\"ahler metrics

In this note we present, for every $n\ge 4$, a non-K"ahler compact complex manifold $X$ of complex dimension $n$ admitting a balanced metric and an astheno-K"ahler metric which is in addition $k$-th Gauduchon for any $1\leq k\leq n-1$.