The analogous logarithmic identities for supersingular lambda invariants at $p=3$

Assume $p=3$. Let $L$ be the set of supersingular lambda invariants in $\mathbf{F}_{N^2}$, let $H'$ be the derivative of the associated polynomial $H$, and extend and lift $\log$ to a surjective group homomorphism

$$\log:\mathbf{F}_{N^2}^{\times}\longrightarrow\mathbf{Z}/p^{r+1}\mathbf{Z}.$$

The $p=3$ logarithmic conjecture. The four assertions in the source hold: existence of $\lambda\in L$ with $\log(\lambda)\not\equiv0\pmod3$, the displayed quadratic-sum congruence modulo $9$, and the two displayed pointwise congruences modulo $3$. The source gives no resolution of these assertions.