Meet obstructions and saturation for the constant window convolution on graded posets

Let $\mathsf{P}$ be a finite graded poset and $Δ_a^{\mathsf{P}}$ the height-$a$ thickening of its diagonal, with projections $q_1,q_2$ to $\mathsf{P}$. We study the \emph{window convolution} $C_a=\operatorname{Lan}_{q_1}\circ q_2^\ast$ on $\mathrm{Shv}(\mathsf{P};k)$, a discrete analogue of convolut