We will prove this using the induction principle of Lemma \ref{lemma-induction-principle}. The property is true for representable quasi-compact and quasi-separated objects of the site $X_{spaces, \etale}$ by Derived Categories of Schemes, Lemma \ref{perfect-lemma-generator-with-support}. \medskip\noindent Assume that $(U \subset X, f : V \to X)$ is an elementary distinguished square such that the lemma holds for $U$ and $V$ is affine. To finish the proof we have to show that the result holds for $X$. Let $P$ be a perfect object of $D(\mathcal{O}_U)$ supported on $T \cap U$ which is a generator for $D_{\QCoh, T \cap U}(\mathcal{O}_U)$. Using Lemma \ref{lemma-lift-map-from-perfect-complex-with-support} we may choose a perfect object $Q$ of $D(\mathcal{O}_X)$ supported on $T$ whose restriction to $U$ is a direct sum one of whose summands is $P$. Write $V = \Spec(B)$. Let $Z = X \setminus U$. Then $f^{-1}Z$ is a closed subset of $V$ such that $V \setminus f^{-1}Z$ is quasi-compact. As $X$ is quasi-separated, it follows that $f^{-1}Z \cap f^{-1}T = f^{-1}(Z \cap T)$ is a closed subset of $V$ such that $W = V \setminus f^{-1}(Z \cap T)$ is quasi-compact. Thus we can choose $g_1, \ldots, g_s \in B$ such that $f^{-1}(Z \cap T) = V(g_1, \ldots, g_r)$. Let $K \in D(\mathcal{O}_V)$ be the perfect object corresponding to the Koszul complex on $g_1, \ldots, g_s$ over $B$. Note that since $K$ is supported on $f^{-1}(Z \cap T) \subset V$ closed, the pushforward $K' = R(V \to X)_*K$ is a perfect object of $D(\mathcal{O}_X)$ whose restriction to $V$ is $K$ (see Lemmas \ref{lemma-pushforward-perfect} and \ref{lemma-pushforward-with-support-in-open}). We claim that $Q \oplus K'$ is a generator for $D_{\QCoh, T}(\mathcal{O}_X)$. \medskip\noindent Let $E$ be an object of $D_{\QCoh, T}(\mathcal{O}_X)$ such that there are no nontrivial maps from any shift of $Q \oplus K'$ into $E$. By Lemma \ref{lemma-pushforward-with-support-in-open} we have $K' =  R(V \to X)_! K$ and hence $$ \Hom_{D(\mathcal{O}_X)}(K'[n], E) = \Hom_{D(\mathcal{O}_V)}(K[n], E|_V) $$ Thus by Derived Categories of Schemes, Lemma \ref{perfect-lemma-orthogonal-koszul-complex} we have $E|_V = Rj_*E|_W$ where $j : W \to V$ is the inclusion. Picture $$ \xymatrix{ W \ar[r]_j & V & Z \cap T \ar[l] \ar[d] \\ V \setminus f^{-1}Z \ar[u]^{j'} \ar[ru]_{j''} & & Z \ar[lu] } $$ Since $E$ is supported on $T$ we see that $E|_W$ is supported on $f^{-1}T \cap W = f^{-1}T \cap (V \setminus f^{-1}Z)$ which is closed in $W$. We conclude that $$ E|_V = Rj_*(E|_W) = Rj_*(Rj'_*(E|_{U \cap V})) = Rj''_*(E|_{U \cap V}) $$ Here the second equality is part (1) of Cohomology, Lemma \ref{cohomology-lemma-pushforward-restriction} which applies because $V$ is a scheme and $E$ has quasi-coherent cohomology sheaves hence pushforward along the quasi-compact open immersion $j'$ agrees with pushforward on the underlying schemes, see Remark \ref{remark-match-total-direct-images}. This implies that $E = R(U \to X)_*E|_U$ (small detail omitted). If this is the case then $$ \Hom_{D(\mathcal{O}_X)}(Q[n], E) = \Hom_{D(\mathcal{O}_U)}(Q|_U[n], E|_U) $$ which contains $\Hom_{D(\mathcal{O}_U)}(P[n], E|_U)$ as a direct summand. Thus by our choice of $P$ the vanishing of these groups implies that $E|_U$ is zero. Whence $E$ is zero.