Bijection between frustrated parking functions and Stirling permutations

Construct a bijection that maps frustrated parking functions of length n with k lucky cars or, equivalently, k lucky spots, to Stirling permutations of order n with k descents.

Background

The paper proves that the number of frustrated parking functions of length n with k lucky cars or spots is the second-order Eulerian number. The same numbers also count descents in Stirling permutations, namely words of length 2n in which each element of [n] occurs twice and every element occurring between the two copies of i is larger than i. The authors explicitly ask whether these two enumerative interpretations can be connected by a direct, natural bijection.