Does the Lovasz Local Lemma imply a single cluster of solutions?

Suppose a CSP instance satisfies the conditions of the Lovasz Local Lemma (LLL). For example, in kSAT where clauses do not repeat variables, we require the maximum degree $D$ is at most $$ D \le \frac{2^k}{ek} $$

The LLL guarantees there is at least $1$ satisfying assignment to this CSP instance. In fact, here the LLL condition implies the existence of exponentially many satisfying assignments.

Can you describe the phase of this CSP instance? For example, do most pairs the satisfying assignments have $o(n)$ Hamming distance? Is there a single cluster of satisfying assignments according to Hamming distance (such as the left-most plot of Figure 2 in KMRSZ06 )?

Relatedly, are there simpler algorithms (such as gradient descent) for CSP instances that satisfy LLL?