Let $\mathbf{C}$ be a Grothendieck site with enough points. Let $p:\mathcal{E}\to \mathcal{F}$ be a map of simplicial presheaves on $\mathbf{C}$. Is it true that $p$ is a local (Kan) fibration if and only if it is a stalkwise fibration?

To fix terminology:

$p$ is called stalkwise fibration if for each point $q$ the map $q^\*(\mathcal{E})\to q^*(\mathcal{B})$ is a fibration.

$p$ is called local fibration if for each $U\in \mathbf{C}$ and each commutative diagram

$\Lambda^n_k\to \mathcal{E}(U)$
| $\phantom{aaaaaa}$|
V $\phantom{aaaaw}$V
$\Delta^n\to \mathcal{B}(U)$
there exists a covering family $(V_i\to U)_i$ such that in the induced square

$\Lambda^n_k\to \Pi_i\mathcal{E}(V_i)$
| $\phantom{aaaaaa}$|
V $\phantom{aaaaw}$V
$\Delta^n\to \Pi_i\mathcal{B}(V_i)$
has a lifting $\Delta^n\to \Pi_i\mathcal{E}(V_i)$

Let $\mathbf{C}$ be a Grothendieck site with enough points. Let $p:\mathcal{E}\to \mathcal{F}$ be a map of simplicial presheaves on $\mathbf{C}$. Is it true that $p$ is a local (Kan) fibration if and only if it is a stalkwise fibration?

To fix terminology:

$p$ is called stalkwise fibration if for each point $q$ the map $q^*(\mathcal{E})\to q^*(\mathcal{B})$ is a fibration.

$p$ is called local fibration if for each $U\in \mathbf{C}$ and each commutative diagram

$$\require{AMScd} \begin{CD} \Lambda^n_k @>>> \mathcal{E}(U)\\ @VVV @VVV \\ \Delta^n @>>> \mathcal{B}(U) \end{CD}$$

there exists a covering family $(V_i\to U)_i$ such that in the induced square

$$\require{AMScd} \begin{CD} \Lambda^n_k @>>> \Pi_i\mathcal{E}(V_i)\\ @VVV @VVV \\ \Delta^n @>>> \Pi_i\mathcal{B}(V_i) \end{CD}$$

has a lifting $\Delta^n\to \Pi_i\mathcal{E}(V_i)$.