The question's in the title and is easily stated, but let me try to give some details and explain why I'm interested. First, a disclaimer: if the answer's not already somewhere in the literature then it could be rather hard; I'm asking this question here because MO is luck enough to have some of the foremost experts on lacunary hyperbolic groups as active participants.

Definitions

1. A group $\Gamma$ is non-Hopfian if there is an epimorphism $\Gamma\to\Gamma$ with non-trivial kernel.

2. A group $\Gamma$ is lacunary hyperbolic if some asymptotic cone of $\Gamma$ is an $\mathbb{R}$-tree.

To motivate this second definition, note that a group is word-hyperbolic if and only if every asymptotic cone is an $\mathbb{R}$-tree.

Lacunary hyperbolic groups were defined and investigated in a paper of Ol'shanskii, Osin and Sapir (although examples of lacunary hyperbolic groups that are not hyperbolic already existed---I believe the first one was constructed by Simon Thomas). They construct examples that exhibit very non-hyperbolic behaviour, including torsion groups and Tarski monsters.

Question

Once again:-

Is there a non-Hopfian lacunary hyperbolic group?

Motivation

My motivation comes from logic, and the following fact.

Proposition: A lacunary hyperbolic group is a direct limit of hyperbolic groups (satisfying a certain injectivity-radius condition).

That is to say, lacunary hyperbolic groups are limit groups over the class of all hyperbolic groups. (Note: the injectivity-radius condition means that there are other limit groups over hyperbolic groups which are not lacunary hyperbolic. I'm also interested in them.) Sela has shown that limit groups over a fixed hyperbolic group $\Gamma$ (and its subgroups) tell you a lot about the solutions to equations over $\Gamma$. For instance, his result that a sequence of epimorphisms of $\Gamma$-limit groups eventually stabilises (which implies that all $\Gamma$-limit groups are Hopfian) has the following consequence.

Theorem (Sela): Hyperbolic groups are equationally Noetherian. That is, any infinite set of equations is equivalent to a finite subsystem.

In the wake of Sela's work we have a fairly detailed understanding of solutions to equations over a given word-hyperbolic group $\Gamma$. But it's still a matter of great interest to try to understand systems of equations over all hyperbolic groups.

Pathological behaviour in lacunary hyperbolic groups should translate into pathological results about systems of equations over hyperbolic groups. A positive answer to my question would imply that the class of hyperbolic groups is not equationally Noetherian. And that would be quite interesting.

Note: This paper of Denis Osin already makes a connection between equations over a single lacunary hyperbolic group and equations over the class of all hyperbolic groups.