Tempo choice is always an interpretation question.
At the 2019 IWBC Gail Williams performed the Hindemith Sonata on her recital, and told a story about what tempo Hindemith wanted.
The story is actually a Farkas story. Hindemith was a frequent (and favorite) guest conductor of the Chicago Symphony, and at one point, riding on the train to Milwaukee for a run out concert, Farkas was chatting with Hindemith and the topic of tempos and his horn sonata came up.
What Hindemith wanted was for all the movements to be at nearly the same base tempo, just with different characters. As printed the first movement is around 100, the second movement around 96, and the last movement is 92-100. Compared to typical tempos, the first movement is often performed a bit faster at around 108, the second movement often slower at around 80, and the last movement I think is typically around 100. With recordings varying, of course.
Tempos of all movements being nearly the same is what Williams aimed for in the IWBC performance, and also in her Summit recording, which was the source of the story — Farkas had heard that she would be recording the work soon, and relayed this important information from Hindemith.
In the case of the IWBC performance, it was a wonderful and intimate performance, avoiding the temptation to be overly aggressive, as I have sometimes heard done. And it was a wonderful conference with a great, supportive atmosphere; if you have the chance to go to an upcoming IWBC you most certainly should.