If $f_0 , f_1$ are two Morse functions defined on some smooth manifold $M$ with a common critical point $p\in M$, the same value $f_0 (p)=f_1 (p)$ and with the same stable discs $W^s_{f_1} (p)=W^s_{f_2} (p)$. Then is it true that there exists an open neighbourhood $U$ of $p$ and a family of Morse functions $f_t$, $0\leq t\leq 1$ connecting $f_0$ and $f_1$? If so, how to prove it?
A generic path between Morse functions will have singularities where the function goes through a birth or death of critical points, or experiencing a "hadle-slide", but here we don't want any bifurcations, and that's where I got stuck.
For those who want references, it's a statement in the proof of a lemma in Section 9.6 of Cieliebak-Eliashberg's book From Stein to Weinstein and Back.
Another update: here's some thoughts I have. By Morse lemma, there exists diffeomorphisms from some open neighbourhood of $p$ to an open subset of $\mathbb{R}^n$ pulling $f_0$ and $f_1$ back to the same standard Morse function near $p$. By composing these two diffeomorphisms we can get a diffeomorphism from $f_0$ to $f_1$, for example, and the question reduces to the following: is the diffeomorphism isotopic to the identity map? This breaks for general diffeomorphisms because the diffeomorphism group of $\mathbb{R}^n$ deformation retracts to $O(n,\mathbb{R})$. However, the above question might be too strong for this question, by which I mean it might suffice for us to isotopy to some set of diffeomorphisms slightly larger than just the identity.