Varieties of strictly n-generated Heyting algebras

For any $n<\omega$ we construct an infinite Heyting algebra $H_n$ which is $(n+1)$-generated but that contains only finite $n$-generated subalgebras. From this we conclude that for every $n<\omega$ there exists a variety of Heyting algebras which contains an infinite $(n+1)$-generated Heyting algebra, but which contains only finite $n$-generated Heyting algebras. For the case $n=2$ this provides a negative answer to a question posed by G. Bezhanishvili and R. Grigolia in [3].