The Whitehead-product conjecture for the derivation bracket

Let $A$ be an $R$-algebra. Write $F_{R\text{-alg}}(A,A)$ for the monoid of $R$-algebra self-maps, $B F_{R\text{-alg}}(A,A)$ for its classifying space, and ${\bf Der}_R(A,A)$ for the derived spectrum of $R$-derivations. The Gerstenhaber bracket on ${\bf Der}_R(A,A)$ induces, via the preceding construction, a Poisson bracket on $\pi_*F_{R\text{-alg}}(A,A)$ for $*>0$. The Whitehead-product conjecture. The bracket described above agrees with the Whitehead product on $B F_{R\text{-alg}}(A,A)$. This asks whether the bracket arising from the derivation or Gerstenhaber structure is the topological Whitehead product on the classifying space; the supplied text states the question but gives no resolution.