The standard derived equivalence characterization for colax dg-category diagrams

Let $I$ be a category, and let $X,X'$ be colax functors from $I$ to $\Bbbk$-linear dg categories. Assume that $X'$ is $\Bbbk$-projective. Write $\mathcal D(X)$ for the derived category associated with the diagram $X$, and write $X'\overset{\mathrm{sd}}{\leadsto}X$ when there exists a quasi-equivalence from $X'$ to a tilting colax functor for $X$.

Standard derived equivalence characterization. The following conditions should be equivalent:

1. There exists an equivalence

$$\mathcal D(X')\to\mathcal D(X).$$

2. There exists a quasi-equivalence from $X'$ to a tilting colax functor for $X$, that is,

$$X'\overset{\mathrm{sd}}{\leadsto}X.$$

This conjecture asks when an equivalence of the derived categories of diagrams is induced by a standard derived equivalence. The preceding problem asks for conditions under which standard derived equivalence is symmetric; the conjecture gives the proposed characterization under the stated $\Bbbk$-projectivity hypothesis.