Generalized Greenberg conjecture for the cyclotomic tower

Let $K$ be a number field, and let $L'_{\infty}$ and $T_{\infty}$ be the fields appearing in the source's Iwasawa-theoretic construction. Generalized Greenberg conjecture. For an arbitrary base field $K$, one has

$$[L'_{\infty}:L'_{\infty}\cap T_{\infty}]<\infty.$$

For a CM field, the source explains that this finiteness is equivalent to the vanishing of $\alpha(X'_{\infty}(-1))^-$ and is related to the finiteness predicted by Greenberg's conjecture. The supplied text gives no resolution status for the stated non-CM generalization.