The mod $p$ Kurihara conjecture for elliptic curves

Fix a prime $p$ and an elliptic curve $E/\mathbb{Q}$ of conductor $N$, with $L$-function coefficients $a_E(n)$, and assume that $p$ is prime to the Manin constant $c_E$ and that $E[p](\mathbb{Q})=0$. Let $\mathcal{N}_{\mathrm{Kato}}=\mathcal{N}_{\mathrm{Kato}}(E;p)$ be the set of squarefree integers that are products of primes $q$ satisfying $(q,N)=1$, $q\equiv 1\text{ \rm mod } p$, and $a_E(q)\equiv 2\text{ \rm mod } p$. For a squarefree integer $Q=q_1\cdots q_r$ with $q_i\in\mathcal{N}_{\mathrm{Kato}}$, define the Kurihara number

$$\delta_Q=\delta_{Q,E,p}:=\sum_{a_1=1}^{q_1-1}\cdots\sum_{a_r=1}^{q_r-1}\left(\prod_{i=1}^r a_i\right)\left\langle\frac{\prod_{i=1}^r(\zeta_{q_i})^{a_i}}{q_1\cdots q_r}\right\rangle_E^+\in\mathbb{Z}_{(p)},$$

where each $\zeta_{q_i}$ is a chosen generator of $(\mathbb{Z}/q_i)^\times$. The mod $p$ Kurihara conjecture. If $E[p](\mathbb{Q})=0$, $p\nmid c_E$, and $p$ does not divide the product of all Tamagawa factors of $E$ over $\mathbb{Q}$, then there exists $Q\in\mathcal{N}_{\mathrm{Kato}}$ such that

$$\delta_Q\not\equiv 0\text{ \rm mod } p.$$

The conjecture asserts the non-vanishing modulo $p$ of a Kurihara number; the cited results of Kim and Burungale--Castella--Grossi--Skinner show that it is true in the majority of cases, while the general statement is not resolved by the supplied context.