Hi guys!! In this question I recently made (mathoverflow.net/questions/390788/…), Dylan Wilson has answered that in case the diagrams are cosimplicial the claim is true. I can't find any reference, do you know one?

That would save me a lot of time... I don't really want to construct all the higher homotopies XD

@AndreaMarino I don't think what you want is true for cosimplicial diagrams: having equivalences on each level commuting up to homotopy with the cosimplicial maps does not guarantee that the homotopy limits are equivalent. What Dylan is saying is that this works for $\mathbb N^{op}$-indexed diagrams, and that in a stable ∞-category cosimplicial diagrams are equivalent to $\mathbb N^{op}$-diagrams, in such a way that the limits correspond (the ref for this is Higher Algebra Section 1.2.4).

However, two cosimplicial diagrams that are equivalent in your weak sense may not have equivalent associated $\mathbb N^{op}$-diagrams.