Forcing shorter answers will definitely reduce their quality. Every token an LLM generates is like a little bit of extra thinking time. Sometimes it needs to work up to an answer. If you end a response too quickly, such as by demanding one-word answers, it's much more likely to produce hallucinations.
Is this proven?
It's certainly true anecdotally. I've seen it personally plenty of times and I've seen it reported plenty of times.
I know Andrej Karpathy mentions it in his youtube series so there's a good chance of it being true.