Stability of 2-class groups in the $\mathbb{Z}_2$-extension of certain real biquadratic fields

Greenberg's conjecture on the stability of $\ell$-class groups in the cyclotomic $\mathbb{Z}_{\ell}$-extension of a real field has been proven for various infinite families of real quadratic fields for the prime $\ell=2$. In this work, we consider an infinite family of real biquadratic fields $K$. With some extensive use of elementary group theoretic and class field theoretic arguments, we investigate the $2$-class groups of the $n$-th layers $K_n$ of the cyclotomic $\mathbb{Z}_2$-extension of $K$ and verify Greenberg's conjecture. We also relate capitulation of ideal classes of certain sub-extensions of $K_n$ to the relative sizes of the $2$-class groups.