The Generalized Pushout Conjecture for infinity-groupoids and controlled theories

Let $X$ be a cofibrant $\infty$-groupoid. For a pushout

\begin{tikzpicture}[node distance=2.5cm] \node(A) {$D^n$}; \node (B) [\right of=A]{$X$}; \node(C)[below of=A]{$D^{n+1}$}; \node (D)[\right of=C]{$X_+$}; \draw[->](B) to node[black,\right=3]{$p$}(D); \draw[->](A) to node {}(B); \draw[->] (A) to node {}(C); \draw[->](C) to node {}(D); \end{tikzpicture}

The Generalized Pushout Conjecture. The following statements hold: first, the map $p:X\to X_+$ is a weak equivalence; second, given a connected diagram of controlled theories $J$, a model $X$ in $\mathbf{Mod}(\mathbf{UGR}(\Gamma))$ with $UX$ cofibrant, and a pushout of diagrams

\begin{tikzpicture}[node distance=2.5cm] \node(A) {$\mathbf{Fr} (D^n)$}; \node (B) [\right of=A]{$X$}; \node(C)[below of=A]{$\mathbf{Fr}(D^{n+1})$}; \node (D)[\right of=C]{$X_+$}; \draw[->](B) to node[black,\right=3]{$p$}(D); \draw[->](A) to node {}(B); \draw[->] (A) to node {}(C); \draw[->](C) to node {}(D); \end{tikzpicture}

the map $p:X\to X_+$ is a weak equivalence. Here $\mathbf{Mod}(\mathbf{UGR}(\Gamma))$ denotes the category of models of the controlled theory associated with $\Gamma$, $U$ is the underlying-model functor, and $\mathbf{Fr}$ denotes the relevant free-model functor. These assertions predict that the specified pushouts preserve weak equivalences under the stated cofibrancy and connectedness hypotheses, supporting the existence and transfer of the canonical semi-model structures discussed in the surrounding context. The conjecture remains unresolved in the supplied text.