Khoroshkin–Piontkovski differential-algebraicity conjecture for Koszul symmetric operads

Let $\mathcal{P}$ be a finitely generated Koszul symmetric operad, and let its Hilbert series be the formal power series encoding its arity-wise dimensions.

Khoroshkin–Piontkovski conjecture. The Hilbert series of $\mathcal{P}$ is differential algebraic over $\mathbb{Z}[t]$.

This conjecture extends the rationality question for Koszul algebras to symmetric operads. The source presents it as still open and notes that it is supported by examples, while the paper proves the corresponding claim for the classified Koszul symmetric set-operads generated by one element of arity $2$.