The nontrivial-invariant conjecture for discrete-time $\mathfrak{sl}_{2}(\mathbb{R})$ coalgebra systems

Let $d>3$ and let $f$ be the function appearing in the discrete-time $\mathfrak{sl}_{2}(\mathbb{R})$ coalgebra system. Let $C$ denote its Casimir function. An invariant is called trivial if it is a linear combination of powers of $C$ of degree at most $\lfloor d/2\rfloor$. Nontrivial-invariant conjecture. If $f$ is different from the functions in the formula for the three known cases, then only trivial invariants are possible:

$$I_{\mathrm{triv}}(t)=\sum_{k=1}^{\lfloor d/2\rfloor}a_kC^k.$$

This conjecture extends the direct calculations for $d=4,5$, where no different integrable systems were found. Its status for general $d>3$ is not established.