Surjectivity of the cyclotomic Milnor-to-Quillen K-theory map

Let $p$ be an odd prime satisfying Vandiver's conjecture. Let $K={\mathbb Q}(\mu_p)$, and let $\mathcal{O}_K$ denote the ring of $p$-integers in $K$. Consider the natural map

$$K_2^M(\mathcal{O}_K)\otimes {\mathbb Z}_p\longrightarrow K_2(\mathcal{O}_K)\otimes {\mathbb Z}_p.$$

Surjectivity conjecture. The natural map is surjective. This conjecture concerns whether Milnor K-theory accounts for the relevant $p$-adic Quillen K-theory under Vandiver's conjecture.