The algebraic trivial-zero conjecture for Artin representations

Let $H$ be the number field occurring in the Artin representation $\rho$, let $\mathbb{Q}_\infty$ be the cyclotomic $\mathbb{Z}_p$-extension of $\mathbb{Q}$, and let $L_p^{\operatorname{alg}}(\rho,\rho^+;T)$ denote the algebraic $p$-adic $L$-function, with $e(\rho,\rho^+)$ the associated trivial-zero exponent. If

$$H\cap\mathbb{Q}_\infty=\mathbb{Q},$$

then the algebraic trivial-zero conjecture.

$$\operatorname{ord}_{T}L_p^{\operatorname{alg}}(\rho,\rho^+;T)=e(\rho,\rho^+).$$

This is presented as the algebraic analogue of a conjectured trivial-zero phenomenon for analytic $p$-adic $L$-functions. The supplied text does not state whether the assertion is known or remains open.