Lindahl's linearizability conjecture for positive-characteristic polynomials

Let $p\geq 3$ be a prime number, let $\mathcal{K}$ be a complete non-Archimedean field of characteristic $p$, and let $\lambda\in\mathcal{K}$ be neither a root of unity nor satisfy $|1-\lambda|\geq 1$. For a polynomial

$$f(z)=\lambda z+\sum_{i=2}^n a_{i-1}z^{i}\in\mathcal{K}[z],$$

Lindahl's conjecture. The polynomial $f$ is linearizable if and only if $a_i=0$ for every $i\geq 2$ such that $p\nmid i$. This conjecture refines Lindahl's known examples of non-linearizable and linearizable power series in positive characteristic, and predicts that the absence of terms whose exponents are not divisible by $p$ exactly characterizes linearizability in this polynomial family.