Are the representations attached to nonclassical $p$-adic eigenforms of weight 1 Hodge-Tate?

Question

Let $f$ be a nonclassical finite slope overconvergent $p$ adic eigenform of weight 1, and let $\rho_f$ be the associated $p$-adic Galois representation of $Gal(\overline{\mathbb{Q}}_p/\mathbb{Q}_p)$. Is $\rho_f|_{Gal(\overline{\mathbb{Q}}_p/\mathbb{Q}_p)}$ necessarily Hodge-Tate? When $f$ is classical or when the weight is larger than 1 this is always true, but in weight $1$ this can perhaps fail because the Sen operator may not be semisimple.