Representation of the diagonal

The following conjecture, if true, would imply the Erdos discrepancy conjecture.

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For all [math]\displaystyle{ C \gt 0 }[/math] there exists a diagonal matrix with trace at least [math]\displaystyle{ C }[/math] that can be expressed as [math]\displaystyle{ \sum_i \lambda_i P_i \otimes Q_i }[/math], where [math]\displaystyle{ \sum_i | \lambda_i | = 1 }[/math] and each [math]\displaystyle{ P_i }[/math] and [math]\displaystyle{ Q_i }[/math] is the characteristic function of a HAP.

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