Conjectural logarithmic identities for supersingular lambda invariants

Assume $p\geq5$. 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 $\log$ to a surjective group homomorphism

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

Conjectural logarithmic identities. The five assertions in the source hold: the two stated quadratic sum identities, the existence of $\lambda\in L$ with $\log(\lambda)\ne0$, and, under $\sum_{k=1}^{(N-1)/2}k\log(k)=0$, the two pointwise identities involving $\log(\lambda'-\lambda)\log(H'(\lambda'))$ and $\log(\lambda'-\lambda)\log(\lambda')$. These assertions were numerically checked for $N<1000$; the source states that part (iii) is proved under the displayed vanishing hypothesis, while parts (iv) and (v) are motivated by refined $\mathscr{L}$-invariant theory.