The converse of the extension property proposition for diagram operads

Let $S$ be a diagram, let $D$ be a subdiagram, and let ${\cal P}$ be an operad. An $(S,\underline D)$-diagram of ${\cal P}$-algebras is an $S$-diagram of differential graded vector spaces in which the vertices of $D$ carry ${\cal P}$-algebra structures and the arrows in $D$ are homomorphisms of ${\cal P}$-algebras. The couple $(S,D)$ has the ${\cal P}$-extension property if the canonical map

$$\phi:{\cal P}_{(S,\underline D)}\to{\cal P}_{\underline S}$$

has a left inverse $\psi:{\cal P}_{\underline S}\to{\cal P}_{(S,\underline D)}$ satisfying $\psi\circ\phi=1\!\!1$.

Extension property converse. If every $(S,\underline D)$-diagram of ${\cal P}$-algebras can be extended to an $S$-diagram of ${\cal P}$-algebras, then the couple $(S,D)$ has the ${\cal P}$-extension property.

The preceding proposition proves that the extension property is sufficient for extending every relative diagram; this conjecture asserts that the condition is also necessary.