Is $St(n,R) \rightarrow St(n+1,R)$ injective?

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For every (not necessarily commutative) ring R and every $n\geq 3$, one can define the n-th (unstable) Steinberg group St(n,R) as in https://ncatlab.org/nlab/show/Steinberg+group. Is the canonical map $St(n,R) \rightarrow St(n+1,R)$ injective?