The non-CM p-adic L-function conjecture for elliptic curves

Let $E/\mathbb Q$ be an elliptic curve and let $p\geq5$ be a prime at which $E$ has good ordinary reduction. With $F_{\infty}=\mathbb Q(E_{p^{\infty}})$ and $G=\operatorname{Gal}(F_{\infty}/\mathbb Q)$, define the periods $\Omega_+(E)$ and $\Omega_-(E)$, the imprimitive value $L_R(E,\rho,1)$, the Euler factors $P_p(\rho,T)$, the unit root $u$ and the other root $w$ of $1-a_pX+pX^2$, and let $f_\rho$ be determined by $p^{f_\rho}$ equal to the $p$-part of the conductor of $\rho$.

The non-CM p-adic L-function conjecture. There exists ${\cal L}_E\in K_1(\Lambda(G)_{S^*})$ such that ${\cal L}_E(\rho)\ne\infty$ for every Artin representation $\rho$ of $G$, and

$${\cal L}_E(\rho)=\frac{L_R(E,\rho,1)}{\Omega_+(E)^{d^+(\rho)}\Omega_-(E)^{d^-(\rho)}}\,e_p(\rho)\,\frac{P_p(\hat\rho,u^{-1})}{P_p(\rho,w^{-1})}\,u^{-f_\rho}.$$

For complex multiplication this conjecture is stated to be true via the two-variable $p$-adic $L$-function. Without complex multiplication, the paper reports only fragmentary numerical evidence, so the general claim remains open.