Stable diffeomorphism implies diffeomorphism for homology 4-spheres

Let $L=L_1\cup L_2$ satisfy the hypotheses of the preceding Kirby-calculus formulation, except that $L_2$ is required only to generate $H_1(S^3-L_1;\mathbb Z)$ rather than $\pi_1(S^3-L_1)$. Suppose that $L'$ is obtained from $L$ by moves (1)--(4) and their inverses, with equally many occurrences of move (4) and its inverse. Stable-destabilization conjecture. The link $L'$ can also be obtained from $L$ using only moves (1)--(3). Equivalently, stable diffeomorphism should imply diffeomorphism for the corresponding integral homology 4-spheres. The source notes that this would imply that homology 4-spheres admit no exotic smooth structures, but calls the stronger assertion probably dubious.