  In this article, we consider joint returns to zero of $n$ Bessel processes
($n\geq 2$): our main goal is to estimate the probability that they avoid
having joint returns to zero for a long time. More precisely, considering $n$
independent Bessel processes $(X_t^{(i)})_{1\leq i \leq n}$ of dimension
$\delta \in (0,1)$, we are interested in the first joint return to zero of any
two of them: \[ H_n := \inf\big\{ t>0, \exists 1\leq i <j \leq n \text{ such
that } X_t^{(i)} = X_t^{(j)} =0 \big\} \,. \] We prove the existence of a
persistence exponent $\theta_n$ such that $\mathbb{P}(H_n>t) =
t^{-\theta_n+o(1)}$ as $t\to\infty$, and we provide some non-trivial bounds on
$\theta_n$. In particular, when $n=3$, we show that $2(1-\delta)\leq \theta_3
\leq 2 (1-\delta) + f(\delta)$ for some (explicit) function $f(\delta)$ with
$\sup_{[0,1]} f(\delta) \approx 0.079$.
