A pathology of asymptotic multiplicity in the relative setting

We point out an example of a projective family $\pi :X\to S$, a $\pi$-pseudoeffective divisor $D$ on $X$, and a subvariety $V\subset X$ for which the asymptotic multiplicity ${\sigma }_V(D;X\mathrm{/}S)$ is infinite. This shows that the divisorial Zariski decomposition is not always defined for pseudoeffective divisors in the relative setting.