The Whitehead-product kernel conjecture for shrinking wedges

Let $n\geq 2$ and let

$$X=\bigcurlyvee_{k\in\mathbb{N}}X_k$$

be a shrinking wedge of finite $(n-1)$-connected CW-complexes. Let $\sigma:X\to\prod_{j=1}^{\infty}X_j$ be the inclusion map. Whitehead-product kernel conjecture. Then

$$\mathcal{W}_{2n-1}(X)=\ker\bigl(\sigma_{\#}:\pi_{2n-1}(X)\to\pi_{2n-1}(\prod_{j=1}^{\infty}X_j)\bigr).$$

The conjecture asserts that every element of $\pi_{2n-1}(X)$ that becomes trivial in the product is accounted for by the subgroup generated by the relevant Whitehead products. The preceding discussion indicates that this is expected for shrinking wedges of finite $(n-1)$-connected CW-complexes, but that new deformation methods are needed to prove it in general.