Order-one differential-algebraicity conjecture for binary-generated Koszul symmetric operads

Let $\mathcal{P}$ be a Koszul symmetric operad generated by one element in arity $2$, and let $f_{\mathcal{P}}$ denote its Hilbert series. Being differential algebraic of order $1$ means that a differential-algebraic relation involves only $f_{\mathcal{P}}$ and its first derivative $f_{\mathcal{P}}'$.

Order-one differential-algebraicity conjecture. The Hilbert series of $\mathcal{P}$ is differential algebraic of order $1$ over $\mathbb{Z}[t]$.

Equivalently, $f_{\mathcal{P}}$ and $f_{\mathcal{P}}'$ are algebraically dependent over $\mathbb{Z}[t]$. The conjecture is motivated by the known examples of Koszul symmetric operads generated by one binary operation, whose Hilbert series satisfy differential-algebraic identities of order $1$; its general status is not specified in the supplied text.