  In the Proper Interval Vertex Deletion problem (PIVD for short), we are given
a graph $G$ and an integer parameter $k>0$, and the question is whether there
are at most $k$ vertices in $G$ whose removal results in a proper interval
graph. It is known that the PIVD problem is fixed-parameter tractable and
admits a polynomial but "unreasonably" large kernel of $O(k^{53})$ vertices. A
natural question is whether the problem admits a polynomial kernel of
"reasonable" size. In this paper, we answer this question by deriving an
$O(k^7)$-vertex kernel for the PIVD problem. Our kernelization is based on
several new observations and a refined analysis of the kernelization.
