Greenberg's algebraic rank conjecture for families

Let $\mathcal X$ be sufficiently small, let $W(f)$ be the global root number of the family, and let ${}^{\rm c}V_\mathcal X$ and ${}^{\rm c}V'_\mathcal X$ be the relevant critical Galois families with Greenberg-style Selmer complexes $\mathbf R^1\boldsymbol\Gamma({}^{\rm c}V_\mathcal X,{}^{\rm c}\mathbf D_\mathcal X)$ and $\mathbf R^1\boldsymbol\Gamma({}^{\rm c}V'_\mathcal X,{}^{\rm c}\mathbf D'_\mathcal X)$. Greenberg's algebraic rank conjecture. One has

$$\operatorname{rank}_{\mathcal O_\mathcal X}\mathbf R^1\boldsymbol\Gamma({}^{\rm c}V'_\mathcal X,{}^{\rm c}\mathbf D'_\mathcal X)=\operatorname{rank}_{\mathcal O_\mathcal X}\mathbf R^1\boldsymbol\Gamma({}^{\rm c}V_\mathcal X,{}^{\rm c}\mathbf D_\mathcal X)= \begin{cases} 1&\text{if }W(f)=-1,\\ 0&\text{otherwise.} \end{cases}$$

This is the algebraic counterpart of Greenberg's generic analytic rank prediction and is proposed using control theorems and Bloch--Kato conjectures.