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The phrase "are rigorously supported" is correct and usable in written English.
It can be used when discussing evidence, arguments, or theories that are strongly backed by data or research.
Example: "The findings of this study are rigorously supported by extensive empirical evidence."
Alternatives: "are strongly backed" or "are firmly established."
Dictionary
are rigorously supported
adverb
In a rigorous manner.
synonyms
Exact(1)
There are, however, substantial concerns about whether the main conclusions are rigorously supported that would need to be addressed before it could be accepted.
Similar(58)
Although they are rigorously supporting informational privacy, they are setting the defaults based not on what's best for their users but on what's best for them.
However, even though her discoveries were rigorously supported by experimental data it took much longer for McClintock's findings to gain acceptance than it had for Darwin's theory.
In the ACLU lawsuit, we ask that all those responsible for public education use research-based methods that are rigorously administered by well trained and supported professionals.
It is a theory rigorously supported by abundant evidence.
Social media are rigorously censored.
But, otherwise, the characters are rigorously anonymous.
Body language and gestures are rigorously documented.
I'm fairly sure book sales are rigorously audited.
We are rigorously applying the rule," said a spokeswoman.
Properties of NRG are rigorously derived.
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Since I tried Ludwig back in 2017, I have been constantly using it in both editing and translation. Ever since, I suggest it to my translators at ProSciEditing.

Justyna Jupowicz-Kozak
CEO of Professional Science Editing for Scientists @ prosciediting.com