Strictification conjecture for weak sections of a left Quillen presheaf

Let $Y$ be a small category and let $\mathbf{M}$ be a left Quillen presheaf on $Y$. Suppose that $\mathbf{M}$ satisfies hypothesis (o), introduced in Section 19, which in particular says that each $\mathbf{M}(y)$ admits all small limits. Say that the weak sections of $\mathbf{M}$ over $Y$ strictify when the morphism

$$Sect(Y,\int_Y\mathbf{M})\longrightarrow Sect(Y,\int'_Y L(\mathbf{M}))$$

is essentially surjective. Strictification conjecture. The weak sections of $\mathbf{M}$ over $Y$ strictify in this sense. The text proves the corresponding assertion for a Reedy category and leaves the extension to an arbitrary category, under hypothesis (o), as a conjecture.