Greenberg's algebraic-rank conjecture for Coleman families

Let $\operatorname{\boldsymbol f}_U$ be a Coleman family, let $r_{\mathrm{min}}(\operatorname{\boldsymbol f}_U)$ be its minimal admissible generic rank, and let $r_{\mathrm{alg},\mathfrak P}(f_k^\flat/\mathds Q)$ denote the algebraic $\mathfrak P$-rank of the even-weight specialization $f_k^\flat$ at an admissible $k\in U\cap\mathscr W_{N,\mathrm{cl}}^0$. Greenberg's algebraic-rank conjecture. The equality

$$r_{\mathrm{alg},\mathfrak P}\bigl(f_k^\flat/\mathds Q\bigr)=r_{\mathrm{min}}\bigl(\operatorname{\boldsymbol f}_U\bigr)$$

holds for all but finitely many admissible $k\in U\cap\mathscr W_{N,\mathrm{cl}}^0$. This is proposed by combining Greenberg's analytic-rank conjecture with the Birch--Swinnerton-Dyer and Beilinson--Bloch--Kato conjectures. The paper presents it as a natural conjecture, without establishing it in the supplied text.