  Katok conjectured that every $C^{2}$ diffeomorphism $f$ on a Riemannian
manifold has the intermediate entropy property, that is, for any constant $c
\in[0, h_{top}(f))$, there exists an ergodic measure $\mu$ of $f$ satisfying
$h_{\mu}(f)=c$. In this paper we consider a conditional intermediate metric
entropy property and two conditional intermediate Birkhoff average properties
for flows. For a basic set $\Lambda$ of a flow $\Phi$ and two continuous
function $g,$ $h$ on $\Lambda,$ we obtain
$$\mathrm{Int}\left\{h_{\mu}(\Phi):\mu\in \mathcal{M}_{erg}(\Phi,\Lambda)\text{
and }\int g d\mu=\alpha\right\}=\mathrm{Int}\left\{h_{\mu}(\Phi):\mu\in
\mathcal{M}(\Phi,\Lambda) \text{ and }\int g d\mu=\alpha\right\},$$
$$\mathrm{Int}\left\{\int g d\mu:\mu\in \mathcal{M}_{erg}(\Phi,\Lambda)\text{
and }h_{\mu}(\Phi)=c\right\}=\mathrm{Int}\left\{\int g d\mu:\mu\in
\mathcal{M}(\Phi,\Lambda) \text{ and }h_{\mu}(\Phi)=c\right\}$$ and
$$\mathrm{Int}\left\{\int h d\mu:\mu\in \mathcal{M}_{erg}(\Phi,\Lambda)\text{
and }\int g d\mu=\alpha\right\}=\mathrm{Int}\left\{\int h d\mu:\mu\in
\mathcal{M}(\Phi,\Lambda) \text{ and }\int g d\mu=\alpha\right\}$$ for any
$\alpha\in \left(\inf_{\mu\in \in \mathcal{M}(\Phi,\Lambda) }\int g d\mu, \,
\sup_{\mu\in \in \mathcal{M}(\Phi,\Lambda) }\int g d\mu\right)$ and any $c\in
(0,h_{top}(\Lambda)).$ In this process, we establish 'multi-horseshoe'
entropy-dense property and use it to get the goal combined with conditional
variational principles. We also obtain same result for singular hyperbolic
attractors.
