This is a follow-up of the question

Is there a bound on the length of the longest Morse trajectory?.

Consider the following setup. Let $(M,g)$ be a (complete) Riemannian Hilbert manifold and $f$ a Palais-Smale Morse function which is bounded below. In the cited question Pietro Majer essentially states that: Let $a$ be a real number. Then, for every critical point $x$ of $f$ there is a neighborhood $U_x$ of $x$ and a constant $C,$ s.t. whenever $\gamma:[0,T]\rightarrow M$ is a Morse trajectory (i.e. $\dot{\gamma}(t)=-\nabla f(\gamma(t))$) with $f(\gamma(0))\leq a$ and $\gamma([0,T])\subseteq U_x$, then the length of $\gamma$ is bounded by $C$.

When Pietro's answer came online, I thought I had a proof for the statement above. Unfortunately I found a gap in my argument which I could not fix. Any ideas?

This is a follow-up of the question

Is there a bound on the length of the longest Morse trajectory?.

Consider the following setup. Let $(M,g)$ be a (complete) Riemannian Hilbert manifold and $f$ a Palais-Smale Morse function which is bounded below. In the cited question Pietro Majer essentially states that: Let $a$ be a real number. Then, for every critical point $x$ of $f$ there is a neighborhood $U_x$ of $x$ and a constant $C,$ s.t. whenever $\gamma:[0,T]\rightarrow M$ is a Morse trajectory (i.e. $\dot{\gamma}(t)=-\nabla f(\gamma(t))$) with $f(\gamma(0))\leq a$ and $\gamma([0,T])\subseteq U_x$, then the length of $\gamma$ is bounded by $C$.

When Pietro's answer came online, I thought I had a proof for the statement above. Unfortunately I found a gap in my argument which I could not fix. Any ideas?