Profinite rigidity of direct products of finitely generated free groups among residually free groups

Let $F$ and $F'$ be finitely generated free groups. A group is residually free if every nontrivial element survives under some homomorphism to a free group.

Direct-product profinite rigidity conjecture. If a finitely presented residually free group $\Gamma$ has the same finite quotients as $F\times F'$, then

$$\Gamma\cong F\times F'.$$

The conjecture asks for profinite rigidity only among finitely presented residually free groups. The surrounding discussion emphasizes that finite generation and finite presentation can differ substantially for direct products of free groups, so this restricted formulation is motivated by known Grothendieck-pair phenomena.