Dydak's compactification conjecture in extension theory

Let $K$ be a countable CW complex, and let $AE(X)$ mean that every map from a closed subset of $X$ to $K$ extends over $X$. A metrizable compactification $\widetilde{X}$ is a compact metrizable space containing $X$ as a dense subspace. Dydak's compactification conjecture. Any separable metrizable space $X$ with $K\in AE(X)$ admits a metrizable compactification $\widetilde{X}$ with $K\in AE(\widetilde{X})$ if and only if $K$ is homotopy dominated by a finite CW complex. The paper states that related compactification conjectures fail for some complexes of the form $L\vee S^2$, so the general assertion is not resolved by the supplied excerpt.