Dimension of the Green measure of skew products with dominant exponent in the fibers

Let $f(z,w)=(p(z),q(z,w))$ be a polynomial skew product of $\mathbb{C}^2$ of degree $d\geqslant 2$ and let $μ$ and $μ_p$ be the Green measures associated to $f$ and $p$ respectively. Let $λ_1 > λ_2$ and $λ_p$ be the Lyapunov exponents of $(f,μ)$ and $(p,μ_p)$ respectively. We show that if $λ_2 = λ_p