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DG enhancements of $\ell$-adic derived categories

This question is similar in flavor to Existence of dg realization for 6 functors

Let $X$ be a complex variety and $D(X)$ the bounded derived category of constructible sheaves (the Euclidean topology and field coefficients say) on $X$. Let $\mathcal{L}\in D(X)$. Then there exists a dg-algebra $\mathcal{E}$ whose cohomology is the shifted $Hom$-groups $Hom(L, L[i])$ in $D(X)$ and such that the derived category of (dg-modules of) $\mathcal{E}$ is equivalent to the triangulated subcategory of $D(X)$ generated by $\mathcal{L}$. (Strictly speaking I need some finiteness conditions but I am going to ignore those for now).

Question 1: Does a similar result hold if $D(X)$ is replaced with the corresponding $\ell$-adic `derived category’?

If the answer to 1) is yes, then:

Question 2: Assume that $X$ is defined over a finite field and $\mathcal{L}$ admits a mixed structure (a la Deligne). Then the shifted $Hom$-groups $Hom(L, L[i])$ in the ordinary (= non-mixed derived category) inherit mixed structures. Is it possible to lift the mixed structure to the dg-algebra $\mathcal{E}$ produced by an affirmative answer to 1?

In the circles I run in it is standard to assume that 1) is `morally' true but I have no real clue as to what a proof would look like. Not so sure about 2). They are both true for some particular $\mathcal{L}$ (such as if $\mathcal{L}$ is the direct sum of $IC$-complexes corresponding to a stratification by contractible strata). However I am looking for something more general.

A related question:

Question 3: Does the category of all perverse sheaves on $X$ (say in the $\ell$-adic or complex algebraic setting) have enough projectives (or injectives)?

Question 2 has an analogue in the setting of mixed Hodge modules. I do not know the answer there either (except for some very special cases). My reason for asking it in the $\ell$-adic setting is my failure at answering it in the Hodge setting (which I am considerably more familiar with than the $\ell$-adic setup).

Added later: The answer to Question 3 is no in general (look at $Ext$ with skyscrapers).