Greenberg's nonvanishing conjecture for Hida-family $p$-adic $L$-functions

Assume that $\mathbb{I}=\mathfrak{O}_L[[T]]$, and for each $k\in\mathbf{Z}_{\geqslant 2}$ let $\boldsymbol{f}_k$ be the $p$-stabilized newform obtained by setting $T=(1+p)^{k-2}-1$ in $\boldsymbol{f}$. Let $L_p^{\tt MTT}(\boldsymbol{f}_k,s)$ be the associated $p$-adic $L$-function, and suppose it satisfies

$$L_p^{\tt MTT}(\boldsymbol{f}_k,s)=-wL_p^{\tt MTT}(\boldsymbol{f}_k,k-s),$$

where $w=\pm1$ is independent of $k$ for $k\equiv2\pmod{2(p-1)}$. Let $e\in\{0,1\}$ be such that $-w=(-1)^e$. Greenberg's nonvanishing conjecture.

$$\frac{L_p^{\tt MTT}(\boldsymbol{f}_k,s)}{(s-k/2)^e}\biggr\vert_{s=k/2}\neq0,$$

for all but finitely many $k\in\mathbf{Z}_{\geqslant2}$ with $k\equiv2\pmod{2(p-1)}$. The assertion predicts that the functional-equation-forced central zero has exactly the prescribed order for almost all weights in the indicated congruence class; it is an application of Greenberg's conjecture on the generic order of vanishing of $p$-adic $L$-functions in Hida families.