  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 $\mathrm{GL}_n(F)$,
with $F$ a non-archimedean local field, where $\tau$ runs through all
irreducible supercuspidal representations of $\mathrm{GL}_r(F)$ and $r$ runs
through positive integers. The Jacquet conjecture asserts that it is enough to
take $r=1,2,\ldots,\left[\frac{n}{2}\right]$. Based on arguments in the work of
Henniart and of Chen giving preliminary steps towards the Jacquet conjecture,
we formulate a general approach to prove the Jacquet conjecture. With this
approach, the Jacquet conjecture is proved under an assumption which is then
verified in several cases, including the case of level zero representations.
