A hereditarily $\ell_1$ subspace of $L_1$ without the Schur property

Let $\infty > p_1 > p_2 > \cdots > 1$. We construct an easily determined $1$-symmetric basic sequence in $\Bigl( \sum\limits_{n=1}^{\infty} \oplus \ell_{p_n} \Bigr)_1$, which spans a hereditarily $\ell_1$ subspace without the Schur property. An immediate consequence is the existence of hereditarily $\ell_1$subspaces of $L_1$ without the Schur property.