The nonvanishing conjecture for the Eisenstein elements $e_0^1$ and $e_0^0$

Let $L$ be the set of supersingular lambda invariants, let $M'$ be the free module generated by the symbols $[\lambda]$ for $\lambda\in L$, and let $e_0^0$ and $e_0^1$ be the elements defined in the source in $M'/p^rM'$. Eisenstein-element nonvanishing conjecture. The elements $e_0^1$ and $e_0^0$ are non-zero modulo $p$. The source gives no resolution of this nonvanishing assertion.