Koszulness and freeness conjectures for compatible structures

Let $\mathrsfs{O}$ be an operad, and let $\mathrsfs{O}^2$ denote the operad of two compatible $\mathrsfs{O}$-structures. An operad is Koszul when it has the Koszul property for operadic duality. The conjecture has two parts.

Koszulness and freeness conjecture. If $\mathrsfs{O}$ is a Koszul operad, then:

1. The operad $\mathrsfs{O}^2$ of two compatible $\mathrsfs{O}$-structures is also Koszul.
2. Free $\mathrsfs{O}^2$-algebras are free as $\mathrsfs{O}$-algebras.

The first assertion generalises the main theorem cited in the source, while the second proposes that the paper's freeness result for compatible associative structures extends to arbitrary Koszul operads. The source presents both assertions as conjectural, and no resolution is supplied here.