The twisted exceptional zero conjecture for elliptic curves

Let $E$ be the elliptic curve under consideration, with split multiplicative reduction at $p$, Tate period $q_E$, and modular degree coprime to the prime $\ell$. Let $R$ be the coefficient ring, let $v_R$ be its valuation, and let $\lambda_R$ denote the associated map on modular symbols. For an integer $c ygeq 1$ coprime to $p$, a \prime $\ell \geq 5$ coprime to the modular degree of $E$, and a Dirichlet character $\chi:(\operatorname{\mathbb{Z}}/c\operatorname{\mathbb{Z}})^{\times}\rightarrow \operatorname{\mathbb{C}}_p^{\times}$ satisfying $\chi(p)=1$, the displayed expressions are understood in the tensor product specified below.

The twisted exceptional zero conjecture. For every such $\chi$,

$$v_R(q_E) \left(\sum_{a\in (\mathbb{Z}/pc\mathbb{Z})^{\times}} \chi(a) \otimes \lambda_R(a)\left[\frac{a}{pc}\right]\right) = \lambda_{R}(q_E)\sum_{a\in (\mathbb{Z}/c\mathbb{Z})^{\times}} \left(\chi(a) \otimes \left[\frac{a}{c}\right]\right)$$

in $\operatorname{\mathbb{Z}}[\operatorname{Im}(\chi)] \otimes_{\operatorname{\mathbb{Z}}} R$, where $\operatorname{\mathbb{Z}}[\operatorname{Im}(\chi)]$ is the $\operatorname{\mathbb{Z}}$-algebra generated by the image of $\chi$.

This is a refined exceptional-zero statement for twisted Mazur--Tate elements. The condition $\chi(p)=1$ ensures the exceptional zero in the split multiplicative setting, while the restrictions on $\ell$ are used in the known partial results; allowing arbitrary $\ell\geq 5$ leads to counterexamples. The conjecture is attributed in the source to Mazur and Tate and is not given a resolution here.