Cristian is spot on and consistent with what I know. You should operate under the assumption that you own the case structure, regardless of the firm, but let me clarify what "candidate-led" actually means in a functional interview setting.
Here's the reality: You are expected to drive the case, meaning you must present your framework, state your hypotheses, request data, and manage the clock. However, no modern, high-stakes consulting interview will let you stick perfectly to your script. The interviewer's job is often to act as a stress test—they will purposely introduce a non-sequitur, cut you off before you finish an analysis, or send you down a rabbit hole just to see if you can pivot cleanly and quickly return to your overall structure.
If you treat the case as rigidly candidate-led, you fail the moment the interviewer redirects you. Instead, view the process as candidate-owned structure plus required adaptability. Horváth, especially in European offices, values both technical rigor and commercial agility. Demonstrate ownership by always summarizing where you are, why you are moving to the next step, and—critically—how you are accommodating the interviewer’s interruption without losing sight of the core objective. This integrated approach is what truly separates successful candidates.
All the best with the preparation.