Spoilers: The murder, the frame-up and the client’s motive are revealed below.
Kogoro’s client kills her husband and frames the detective. The Detective Who Lost His Mind is international episode 1184 (Japanese episode 1126), an anime-original case. A matching suit and Kogoro’s fingerprints make the accusation look convincing, but the shadows in the security footage expose a false recording time.
The important distinction is between genuine traces and the story built around them. Kogoro really touched the trophy later used as the weapon; that does not mean he touched it while committing the murder.
An infidelity inquiry gives the client access to Kogoro

Conan and Kogoro visit the home of the Kon Corporation chairman and his wife. Mrs. Kon wants her husband followed because he is fascinated by popular jockey Sakura Sakurada, who rides the company’s racehorse. She presents the request as anxiety about a possible affair.
Kogoro spends a week watching the chairman without finding suspicious conduct. At the house, however, he has already provided his future accuser with something useful: carried away by the racing trophies, he picks one up and leaves his fingerprints on it.
When he gives his report during a Ferris-wheel ride, Mrs. Kon appears relieved and invites him to dinner. She arrives with a strikingly shorter haircut, and Kogoro grows increasingly flattered by her attention. Conan’s presence, at Ran’s insistence, keeps an observer beside a detective whose judgment is being distracted.
The evening ends with a scream at the house

After dinner, Kogoro takes the slightly tipsy Mrs. Kon home. Her scream draws him inside, where her husband lies dead with a fatal injury to the back of his head. A trophy in the room is identified as the weapon.
The living-room security camera seems to provide direct evidence. It shows a masked figure striking the chairman with the trophy, and the attacker wears the same kind of suit as Kogoro. The fingerprints on the weapon make the visual resemblance much harder to dismiss.
Mrs. Kon initially appears to support Kogoro’s alibi, then recalls that they had separated between the Ferris-wheel report and their later meeting. Her visit to a hair salon gives her an apparently safe account of the relevant period, while the gap in his movements leaves him exposed.
Kogoro’s effort to defend his client only makes the position more painful. Megure moves toward taking him into custody, and Conan finally intervenes by putting him to sleep and delivering a Sleeping Kogoro deduction.
The camera’s clock cannot explain the shadows

The security footage carries a time, but the length of the window shadows does not fit that time. Conan identifies the contradiction between the displayed clock and the light visible in the room. The camera setting has been altered to place the killing in the wrong part of the day.
That correction changes which alibis matter. Kogoro was at a barber’s at the actual time of the crime. A small shaving injury, covered by a plaster, helps substantiate that visit. A mundane mark on his face counters a much more dramatic image of someone in his suit.
The deduction does not need to deny what the camera records. It needs to distinguish what is visible from when the camera claims it happened. Once the time is corrected, the apparent alignment between Kogoro’s opportunity and the masked attack breaks apart.
Why the trophy and the suit point to the client’s preparation

Mrs. Kon had seen, through the security recording, that Kogoro handled the trophy. She chose that object as her weapon because it already carried the detective’s prints. She also obtained a matching suit to make the filmed attacker resemble him while hiding her face.
The false timestamp completes the arrangement. The fingerprints suggest contact with the weapon, the suit suggests an identity, and the manipulated time makes Kogoro’s gap in company appear relevant. None of those elements alone supplies the whole accusation; she has made them support one another.
The opening trophy scene therefore has two meanings. At first it is Kogoro enjoying a wealthy client’s home. In retrospect, it is the moment the client discovers a ready-made piece of evidence she can use against him.
The real obsession is the company

Confronted with the broken timeline, Mrs. Kon admits the crime and reveals an ambition very different from the marital worry that brought Kogoro to her. Her attachment is to Kon Corporation. She believes she could make it greater and has come to see her husband, the chairman, as the obstacle.
The jockey and suspected affair provide the entry point to the scheme, but the motive is control of the business. That distinction makes the dinner invitation especially cold: Kogoro reads it as personal appreciation while she is positioning him to take responsibility for her murder.
Afterward, Megure insists he never truly doubted Kogoro, provoking the detective’s skeptical response. A meal restores some familiar comedy. The relief works because the crucial rescue came from details that the glamorous evening encouraged Kogoro to overlook—the trophy, the clock, the shadow and a barber’s small mistake.

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