The stationary law of the leftmost particles in Aldous's drift-jump process

In Aldous's drift-jump process, $k$ particles $0<X_1<\dots<X_k$ move to the right with velocity equal to their position, and at the events of a rate-one space-time Poisson process the nearest particle to the right of the event jumps to it. Aldous asked for a fairly explicit description of the stationary distribution. We give a self-contained proof that the stationary distribution is unique and that in stationarity $X_i=\Gamma_{\tau_i}$, where $\Gamma_1<\Gamma_2<\dots$ are the points of a rate-one Poisson process and, independently, $\tau_i$ is the first $n$ for which the longest increasing subsequence of $n$ independent uniform variables has length $i$; this is a form of a representation of Aldous and Diaconis. Equivalently, $\mathbb P(X_{k_1}>s_1,\dots,X_{k_r}>s_r)$ is the probability that the Poissonized longest increasing subsequence stays below $k_a$ at the times $s_a$. We evaluate every such probability as $e^{-s_r}$ times a finite determinant with Bessel-function entries, of size at most $k(k-1)$ for the first $k$ particles. For a single particle this is Gessel's Toeplitz determinant. As a consequence $\mathbb E(X_i-X_{i-1})$ is the constant $c(i)$ of Aldous and Diaconis in the logarithmic growth of the $i$-th pile in patience sorting.