Meylan–Mir–Zaitsev approximation conjecture for real analytic targets

Let $M\subset\mathbb C^N$ be a real analytic CR submanifold, let $p\in M$ be a point of finite type, and let $K$ be a positive integer. Let $M'\subset\mathbb C^{N'}$ be a real analytic set, let $p'\in M'$, and let $F:(\mathbb C^N,p)\to(\mathbb C^{N'},p')$ be a formal mapping sending $M$ into $M'$. Meylan–Mir–Zaitsev approximation conjecture. The conclusion of the approximation theorem should still hold when $M'$ is assumed to be a real analytic set rather than an algebraic set: there exists a germ of a holomorphic mapping $H:(\mathbb C^N,p)\to(\mathbb C^{N'},p')$ sending $M$ into $M'$ with

$$H(Z)-F(Z)=O(|Z-p|^K).$$

The conjecture removes the algebraicity hypothesis from the known theorem and concerns formal-to-holomorphic approximation for finite-type CR submanifolds.