Consider a cubic threefold $Y$ and its associated degree $14$ prime Fano threefold $X$, we have the equivalences of non-trivial components of $D^b(Y)$ and $D^b(X)$, i.e, $\mathcal{A}_X\cong\mathcal{B}_Y$ and this equivalence is given by a Fourier-Mukai functor $\Phi:=\Phi_{I_Z(H_Y)}:D^b(X)\rightarrow D^b(Y)$, where $Z\subset X\times Y$ is an irreducible four dimensional subvariety. Consider the projection map $\pi:Z\rightarrow Y$, my question is how to describe the fiber of this map $\pi^{-1}(y)$ for every point $y\in Y$, which is $Z\cap X_y$.

Since $X_y\cong X$ is a prime Fano threefold, its Picard number is 1, so it can not contain any divisor which is not multiple of the degree 14 $K3$ surface, by playing some game with matrices, it seems to me that $\pi^{-1}(y)$ is degree 2, so it looks like it is of dimension 1 and degree $2$, which looks like a conic on $X$, but I have difficulty to show this fiber is connected to exclude the possibility that two disjoint lines.