  The systematic study of inversion sequences avoiding triples of relations was
initiated by Martinez and Savage. For a triple
$(\rho_1,\rho_2,\rho_3)\in\{<,>,\leq,\geq,=,\neq,-\}^3$, they introduced
$\I_n(\rho_1,\rho_2,\rho_3)$ as the set of inversion sequences $e=e_1e_2\cdots
e_n$ of length $n$ such that there are no indices $1\leq i<j<k\leq n$ with $e_i
\rho_1 e_j$, $e_j \rho_2 e_k$ and $e_i \rho_3 e_k$. To solve a conjecture of
Martinez and Savage, Lin constructed a bijection between $\I_n(\geq,\neq,>)$
and $\I_n(>,\neq,\geq)$ that preserves the distinct entries and further posed a
symmetry conjecture of ascents on these two classes of restricted inversion
sequences. Concerning Lin's symmetry conjecture, an algebraic proof using the
kernel method was recently provided by Andrews and Chern, but a bijective proof
still remains mysterious. The goal of this article is to establish bijectively
both Lin's symmetry conjecture and the $\gamma$-positivity of the ascent
polynomial on $\I_n(>,\neq,>)$. The latter result implies that the distribution
of ascents on $\I_n(>,\neq,>)$ is symmetric and unimodal.
