Counting algebraic points of bounded height Let $K$ be a number field and $X\hookrightarrow\mathbb P^n_K$ be a projective variety of degree $\delta$ (with respect to universal bundle) and dimension $d$. We denote the set
$$S(X;D,B)=\{\xi\in X(\overline K)|[K(\xi):K]\leq D,\;H_{K(\xi)}(\xi)\leq B\}.$$
and
$$N(X;D,B)=\#S(X;D,B),$$
where $H_{K(\xi)}(\cdot)$ is the common naive height of an algebraic point over the field $K(\xi)$. 
I guess that for all $X$ with degree $\delta$ and dimension $d$, we might have 
$$N(X;D,B)\ll_{n,K,D}\delta N(\mathbb P^d_K;D,B),$$
since for the case of $D=1$, which is the rational points' case, we can prove the result. But I don't know how to prove it. 
If this question is too difficult, at least could we prove the complete intersection case? I can only prove the case of hypersurfaces. 
Thank you very much. 
PS. In this question, the definition of algebraic points of bounded height is different from the usual definition, which is 
$$S(X;D,B)=\{\xi\in X(\overline K)|[K(\xi):K]=D,\;H_{K(\xi)}(\xi)\leq B\}.$$
 A: This is true. Choose any linear subspace $Z$ (over $K$) of dimension $n- d-1$ and disjoint from $X$, and take $\pi : \mathbb{P}_K^n \setminus Z \to \mathbb{P}_K^d$ the corresponding linear projection. Next, consider a linear automorphism $g \in \mathrm{PGL}(d,K)$ such that $g(Z)$ is the standard linear subspace $Z_0$ defined by the vanishing of the last $n-d-1$ coordinates; $g$ of course depends on $X$.
You can use the projection $\pi' := \pi \cdot g : \mathbb{P}_K^n \setminus Z_0 \to \mathbb{P}_K^d$. It is independent of $X$, hence all $P \in \mathbb{P}^n(\bar{K})$ satisfy $H_K(\pi'(P))  \leq C H_K(P) $ with a constant $C$ depending just on $n$.  The original projection, $\pi = \pi' \cdot g^{-1}$, thus sends $\{ P \in X(\bar{K}) \mid [K(P):K] \leq D, \, H_K(P) \leq  B \}$ at most $\delta:1$ into the set $\{ Q \in \mathbb{P}^d(\bar{K}) \mid [K(Q):K] \leq D, \, H_K(g' \cdot Q) \leq C \cdot B \}$, where $g' \in \mathrm{PGL}(d,K)$ is a certain element (induced by $g^{-1}$).  But the last set is in bijection $Q \leftrightarrow g'^{-1} \cdot Q$ with  $\{  Q \in \mathbb{P}^d(\bar{K}) \mid [K(Q):K] \leq D, \, H_K( Q) \leq  C \cdot B \} $, and therefore it has the same cardinality. Finally, the latter cardinality is
$$
\asymp_C \# \{  Q \in \mathbb{P}^d(\bar{K}) \mid [K(Q):K] \leq D, \, H_K( Q) \leq   B \},
$$
as follows for instance from a generalization of Northcott's precise asymptotics to points of a bounded degree, as proved in: [D. Masser, J. Vaaler: Counting algebraic numbers with large height II, Trans. Amer. Math. Soc., 2006.]
