The positivity property for canonical bases asserts that the structure constants of the multiplication for the canonical basis are in ${ \mathbb N}[v,v^{ -1}]$. Let $\mathbf U$ be the quantum group over ${ \mathbb Q}(v)$ associated with a symmetric Cartan datum. The positivity property for the positive part ${ \mathbf U}^+$ of ${ \mathbf U}$ was proved by Lusztig. He conjectured that the positivity property holds for the modified form $\dot{ \mathbf U}$ of ${ \mathbf U}$. In this paper, we prove that the structure constants for the canonical basis of $\dot{ { \mathbf U}}(\widehat{ \frak{ sl}}_n)$ coincide with certain structure constants for the canonical basis of ${ \mathbf U}(\widehat{ \frak{ sl}}_N)^+$ for $n<N$. In particular, the positivity property for $\dot{ { \mathbf U}}(\widehat{ \frak{ sl}}_n)$ follows from the positivity property for ${ \mathbf U}(\widehat{ \frak{ sl}}_N)^+$.
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