Conjectural structure of the Iwasawa module in a real quadratic case

Let $K=\mathbb{Q}(\sqrt{D})$, where $D=pq$ with $p\equiv 5\pmod 8$ and $q\equiv 1\pmod 8$, and suppose

$$\left(\dfrac{2}{q}\right)_4\neq (-1)^{\frac{q-1}{8}}.$$

Assume also

$$\left(\dfrac{p}{q}\right)=1,\qquad \left(\dfrac{p}{q}\right)_4=-\left(\dfrac{q}{p}\right)_4,\qquad q\equiv 9\pmod {16},\qquad N(\varepsilon_{2D})=-1.$$

Let $X_{\infty}$ be the Iwasawa module of the cyclotomic $\mathbb{Z}_2$-extension of $K$, and let $A_1$ denote the class-group module used in the paper. The paper's conjecture. Under these assumptions,

$$X_{\infty}\cong A_1.$$

This is proposed for a case left unresolved in the cited classification. The surrounding discussion says that the module is known to be finite in some comparable cases, but that the point where the class-group chain stops growing is difficult to predict; no resolution of this specific assertion is given.