Let $K$ be a finite extension of $\mathbb{Q}$ and $L/K$ be a $\mathbb{Z}_p$-extension with finite layers $L_i$, hence $L_j/L_i$ is cyclic of order $p^{j-i}$ (put $K=L_0$). Let $U_E$ be the unit group of the ring of integers of the number field $E$. Note that we know after a finite layer, $L_{i+1}/L_{i}$ is totally ramified, and we assume we work in the totally ramified layers. $H^1(Gal(L_{i+1}/L_i),U_{L_{i+1}})$ is an elementary abelian $p$-group since $L_{i+1}/L_i$ is of degree $p$. Now, can we say that cohomology groups $H^1(Gal(L_{i+1}/L_i),U_{L_{i+1}})$ and $H^1(Gal(L_{i+2}/L_{i+1}),U_{L_{i+2}})$ have the same order?
Using the Herbrand quotient, this is equivalent to: can we say that the indices of norms of units (i.e. order of zero$^{th}$ Tate cohomology groups with coefficient in units) in the extensions $L_{i+2}/L_{i+1}$ and $L_{i+1}/L_i$ are equal? Note that we know these indices are powers of $p$, using the Herbrand quotient and unramifiedness of infinite primes in $\mathbb{Z}_p$-extensions.
