Whether you lift lighter weights or heavier weights until you can't do any more reps, both ways give you the same muscle growth and strength gains if you're already used to training.
Evidence from Studies
Supporting (2)
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Neither load nor systemic hormones determine resistance training-mediated hypertrophy or strength gains in resistance-trained young men
This study shows that lifting lighter weights many times or heavier weights fewer times, both until you can't do more, gives similar muscle growth and strength gains in men who already work out.
Divergent Strength Gains but Similar Hypertrophy After Low-Load and High-Load Resistance Exercise Training in Trained Individuals: Many Roads Lead to Rome.
The study shows that lifting lighter weights many times or heavier weights fewer times, both done until failure, give similar muscle growth and strength gains in trained people, just like the claim says.
Contradicting (2)
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Low-Load Resistance Training to Volitional Failure Induces Muscle Hypertrophy Similar to Volume-Matched, Velocity Fatigue
The study didn't test high weights with pushing until failure like the claim says, so it can't fully support it, and it actually shows high weights without failure gave better strength gains.
Effects of Resistance Training Performed to Failure or Not to Failure on Muscle Strength, Hypertrophy, and Power Output: A Systematic Review With Meta-Analysis.
The study looked at training to failure versus not to failure and found that training to failure might lead to more muscle growth, which doesn't match the claim that both low and high loads to failure give the same results.
Gold Standard Evidence Needed
According to GRADE and EBM methodology, here is what ideal scientific evidence would look like to definitively prove or disprove this specific claim, ordered from strongest to weakest evidence.