Finiteness conjecture for filtered lambda-ring structures with prescribed Adams linear coefficients

Let $n$ be an integer with $n\geq 3$, and let

$$\{b_p\in p\mathbf{Z}:p\text{ prime}\}$$

be a family of nonzero integers. A filtered $\lambda$-ring structure on $\mathbf{Z}[x]/(x^n)$ has prescribed Adams linear coefficients $\{b_p\}$ when its Adams operations satisfy

$$\psi^p(x)\equiv b_p x\pmod{x^2}$$

for every prime $p$. Finiteness conjecture. There are only finitely many isomorphism classes of filtered $\lambda$-ring structures on $\mathbf{Z}[x]/(x^n)$ with these prescribed Adams linear coefficients. The case $n=3$ is contained in a theorem, but the assertion for general $n$ remains open.