Diagonal of a diagram of codescent objects Given the following diagram in a $2$-category, in which squares of the same  "type" commute, where each column and each row is a strong codescent diagram (Edit: it should be reflexive as well), is then the diagonal a codescent diagram as well?

Actually, for each row and each column we also have six $2$-isomorphisms (for example, $\xi^h_i : x^h_i d^h_i \to x^h_i c^h_i$ for the $i$th row), which I didn't write down here. 
This question is motivated by Mike Shulman's comment here. In fact, this statement would probably be an important step to construct bicategorical pushouts of symmetric pseudomonoids.
I tried to find a proof, but basically get lost because of this huge amount of data.
Notice that this statement (if it is true) is a generalization of the corresponding $1$-dimensional statement about reflexive coequalizers (Sketches of an elephant, Lemma A.1.2.11). I would be very happy for a reference to the literature where this is proven. Instead of writing down a proof on my own, which will be probably too long anyway, I would like to cite and then use this result in a paper.
 A: As I wrote in a comment above, for this result to hold, each row and column of your diagram must be a reflexive codescent diagram. I do not know of any place in the literature where this result is explicitly stated, but, as I will explain below, it follows without difficulty from a result of Steve Lack.
(For simplicity, let me deal only with strict reflexive codescent objects. Since these are flexible colimits, one can deduce the fully weak bicategorical version of this result from the strict version by standard arguments.) 
Definition. Let $\Delta_{\leq 2}$ denote the full subcategory of the simplex category $\Delta$ containing the objects $[0]$, $[1]$, and $[2]$, and let $W : \Delta_{\leq 2} \longrightarrow \mathbf{Cat}$ denote the composite of the full inclusion $\Delta_{\leq 2} \longrightarrow \mathbf{Cat}$ with the groupoid reflection functor $\mathbf{Cat} \longrightarrow \mathbf{Cat}$.  For any $2$-category $\mathcal{K}$, the reflexive codescent object of a functor $X : \Delta_{\leq 2}^\mathrm{op} \longrightarrow \mathcal{K}$ is the colimit $W \ast X$ of $X$ weighted by $W$.
We will deduce the "diagonal lemma" of your question for reflexive codescent objects from the fact that reflexive codescent objects are sifted colimits (in the $\mathbf{Cat}$-enriched sense), i.e. that the functor $$W \ast (-) : [\Delta_{\leq 2}^\mathrm{op},\mathbf{Cat}] \longrightarrow \mathbf{Cat}$$ preserves finite products. This fact is due to Steve Lack -- see Proposition 4.3 of

Lack, Stephen. Codescent objects and coherence. J. Pure Appl. Algebra 175 (2002), no. 1-3, 223--241. doi

and Proposition 4 of

Bourke, John. A colimit decomposition for homotopy algebras in Cat. Appl. Categ. Structures 22 (2014), no. 1, 13--28. doi

Thanks to the "Fubini theorem" for iterated weighted colimits, we may state the diagonal lemma for reflexive codescent objects in the following form.
Lemma (diagonal lemma for reflexive codescent objects). Let $\mathcal{K}$ be a $2$-category and let $X \colon \Delta_{\leq 2}^\mathrm{op} \times \Delta_{\leq 2}^\mathrm{op} \longrightarrow \mathcal{K}$ be a functor. Then we have an isomorphism of weighted colimits in $\mathcal{K}$
$$W \ast (X \circ \delta) \cong (W \times W) \ast X,$$
either side existing if the other does. (Here $\delta$ denotes the diagonal functor $\Delta_{\leq 2}^\mathrm{op} \longrightarrow \Delta_{\leq 2}^\mathrm{op} \times \Delta_{\leq 2}^\mathrm{op}$).
Remark. It is also worth displaying the isomorphism of this lemma in coend form:
$$\int^{[k]} W^k \times X_{k,k} \cong \int^{[n],[m]} W^n \times W^m \times X_{n,m}.$$
Proof of lemma. The preservation of binary products of representables by the functor $W \ast (-) : [\Delta_{\leq 2}^\mathrm{op},\mathbf{Cat}] \longrightarrow \mathbf{Cat}$ implies, via the weighted colimit formula for left Kan extensions, that the functor $W \times W :  \Delta_{\leq 2} \times \Delta_{\leq 2} \longrightarrow \mathbf{Cat}$ is the left Kan extension of $W \colon \Delta_{\leq 2} \longrightarrow \mathbf{Cat}$ along the diagonal functor $\delta \colon \Delta_{\leq 2} \longrightarrow \Delta_{\leq 2} \times \Delta_{\leq 2}$. Hence the lemma follows from Theorem 4.38 of Kelly's Basic concepts of enriched category theory. $\Box$

It is worth mentioning that, when working bicategorically (i.e. "up to equivalence"), the codescent object of a (pseudo)functor $X : \Delta_{\leq 2}^\mathrm{op} \longrightarrow \mathcal{K}$ is simply its bicolimit. Hence the bicategorical version of the diagonal lemma for reflexive codescent objects -- which follows from the strict version by standard arguments -- is simply the statement that the diagonal functor $\Delta_{\leq 2}^\mathrm{op} \longrightarrow \Delta_{\leq 2}^\mathrm{op} \times \Delta_{\leq 2}^\mathrm{op}$ is 2-final. 
