Undecidability of epimorphisms onto products of hyperbolic groups

We exhibit examples of finitely presented subgroups $P$ of direct products of hyperbolic groups for which there is no algorithm that detects whether a finitely presented group has a quotient isomorphic to $P$. For any torsion-free, linear, hyperbolic group $Q$ that maps onto the free group of rank $2$ and $m\ge 2$, we construct a recursive sequence $({\mathrm{\Gamma }}_n{)}_{n\in \mathbb{N}}$ of torsion-free, hyperbolic $C^{\mathrm{\prime }}(\frac{1}{6})$ small cancellation groups, with the property that there is no algorithm determining the values $n\in \mathbb{N}$ such that ${\mathrm{\Gamma }}_n$ has a quotient isomorphic to the direct product $Q^m$ of $m$-copies of $Q$.