Almost-complex-to-complex conjecture for closed manifolds

For every natural number $n$ satisfying $3\le n$ and every type $M$ in an arbitrary universe, equipped with a topology, a charted-space structure modeled on $\mathbb R^{2n}$, and a $C^\infty$ real manifold structure for those charts, assume that $M$ is Hausdorff, second countable, compact, and connected, where connectedness includes nonemptiness. For every supplied family of continuous real-linear maps $J_x:T_xM\to T_xM$ satisfying $J_x(J_x(v))=-v$ for all $x\in M$ and $v\in T_xM$, and such that $(x,v)\mapsto(x,J_x(v))$ is a $C^\infty$ map of the given real tangent bundle to itself, there exists a charted-space structure on the same topological space $M$, modeled on $\mathbb C^n$, such that all three conditions hold: the resulting manifold is $C^1$ over $\mathbb C$; the identity map from $M$ with its original real charts to $M$ with the new complex charts, regarded as charts into the real vector space underlying $\mathbb C^n$, is $C^\infty$ over $\mathbb R$; and the identity map in the reverse direction is also $C^\infty$ over $\mathbb R$. Existence, not uniqueness, of the new charted-space structure is asserted. No equation or other compatibility condition between the supplied maps $J_x$ and multiplication by $i$ in the new charts is required. The dimension hypothesis excludes $n=0,1,2$, and the connectedness assumption excludes an empty $M$.