Finiteness conjecture for topologically realizable filtered lambda-ring structures

Let $n$ be an integer with $n\geq 3$. Consider the filtered truncated polynomial ring

$$\mathbf{Z}[x]/(x^n).$$

A filtered $\lambda$-ring structure is a filtered $\lambda$-ring structure on this ring, and a structure is topologically realizable if it is isomorphic to the $K$-theory of a torsionfree space. Finiteness conjecture. Among the uncountably many isomorphism classes of filtered $\lambda$-ring structures on $\mathbf{Z}[x]/(x^n)$, only finitely many are topologically realizable. The cases $n=3$ and $n=4$ are known, while the assertion for general $n$ remains open.