Which of the following would most likely cause a transmission to stay in first gear until very high engine speed?

A transmission that remains in first gear until high engine speeds is typically a result of a stuck or restricted governor valve. The governor in an automatic transmission is responsible for regulating shift points based on the vehicle's speed. If the governor valve becomes stuck or restricted, it fails to send the appropriate signals or pressure to the transmission, preventing it from shifting into higher gears at the expected speeds. This means the transmission may remain in first gear longer than it should, causing the engine to reach higher RPMs before shifting.

In contrast, a faulty throttle position sensor could affect engine performance and shifting characteristics, but it would not specifically cause the transmission to remain in first gear. Instead, it might lead to erratic shifting or improper shift timing overall. Malfunctioning torque converters can create issues related to power transfer and may affect shifting, but their primary role is not directly related to the governor's function. Worn transmission bands can contribute to slipping or failure to engage gears properly, but they are less likely to exclusively cause the symptom of remaining in first gear until high RPMs.