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The Local Converse Problem is to determine how the family of the local gamma factors $\gamma (s,\pi \times \tau ,\psi )$ characterizes the isomorphism class of an irreducible admissible generic representation $\pi$ of ${\mathrm{GL}}_n\left(F\right)$, with $F$ a non-archimedean local field, where $\tau$ runs through all irreducible supercuspidal representations of ${\mathrm{GL}}_r\left(F\right)$ and $r$ runs through positive integers. The Jacquet conjecture asserts that it is enough to take $r=1,2,...,\left[\frac{n}{2}\right]$. Based on arguments in the work of Henniart and of Chen giving preliminary steps towards the Jacquet conjecture,...