What would allow the engine to start in any gear on a vehicle equipped with an auto-shift transmission?

In an auto-shift transmission, starting the engine while the vehicle is in gear is typically prevented for safety reasons. The gear indicator display plays a crucial role in the starting process by providing information about the current gear selection. If this display is defective, it may not accurately show that the vehicle is in a safe starting position (like neutral), potentially allowing the engine to start while in gear. This situation could lead to unintended vehicle movement, which compromises safety.

Conversely, the other scenarios mentioned involve mechanical or electrical issues that wouldn't inherently allow the engine to start in gear. A stuck shift finger or a malfunctioning gear select motor typically take an active role in the shifting process, and a defective start enable relay would not affect the indication of the gear selection itself but rather the engine starting process overall. These factors do not create the same opportunity for starting in gear that a defective gear indicator display does, which is why the latter is the most plausible choice in this scenario.