McCallum–Sharifi conjecture on cyclotomic p-unit cup products

Let $p$ be a prime, let $K=\mathbb{Q}(\mu_p)$, and let $C_K$ be the subgroup of cyclotomic $p$-units generated by $-1$ and the elements $\zeta_p^i-1$. The cup-product pairing

$$\langle\cdot,\cdot\rangle_p:C_K\times C_K\longrightarrow A_K^-\otimes\mu_p$$

is defined using the Kummer pairing, where $A_K$ is the $p$-primary part of the class group of $K$.

McCallum–Sharifi conjecture. For all primes $p$, the pairing $\langle\cdot,\cdot\rangle_p$ is nontrivial.

This is a rephrasing of McCallum and Sharifi's Conjecture 5.3 concerning cup products of cyclotomic $p$-units and the minus part of the $p$-primary class group. The supplied source does not state a resolution, so the conjecture remains open.