Okay, got some sweet string memes brewing:
Just kidding.
It seems likely the mystery man sneaking around the mansion was Rudolf himself. He found someway to get up to the window, and had actually smuggled in the trenchcoat/hat instead of food. He jumped from the window, explaining the bruises found. However, he still walked it off and snuck into the mansion via a rope hanging from the dining room window. From there, he sneaks into Kinzo’s study with the intent of finding the will. The whole premise here is that the children have no idea what Kinzo intends to actually do if he carries out his suicide. So, Rudolf wants to have all the information and then decide whether to stay in the tower (forcing Kinzo’s will to be enacted), or to give up, avoiding the will’s contents.
He finds the will, and runs into Kanon, who chases him upstairs. Rudolf escapes via the hallway window upstairs, grabbing onto the rope hanging out of the dining room window directly below (hence the undisturbed mud).
From here, things get hazy. Knowing the contents of the will are beneficial for all the siblings, Rudolf has no reason to murder anyone. Krauss is suspicious since he might have been stashing the wet trenchcoat/hat in the storage room. If the culprit is not Kinzo, then it is likely the killer waited for Gohda to leave his post, then quickly slip into the passage, kill Kinzo, take the key from him, and leave, sealing the passage behind them. My money is on Krauss right now, he’s the only one who the purple statements would allow to slip away after Rudolf’s discovery (and hence Gohda leaving his post) to kill Kinzo.
Thinking about it a little more, killing Nanjo might be to point the blame towards Genji. Him being out of the picture gives Genji the whole mansion to himself, and we know that Genji was one of the few who knew the contents of the will in the first place.