Noncommutative Tao uncertainty principle for prime dimensions

Let $p$ be a prime and let $\mathcal{A}$ be a unital C*-algebra with the invariant basis number property. Define the noncommutative Fourier transform $\widehat{\phantom{x}}:\mathcal{A}^p\to\mathcal{A}^p$ by

$$\widehat{(a_j)_{j=0}^{p-1}}=\frac{1}{\sqrt{p}}\left(\sum_{j=0}^{p-1}a_j e^{\frac{-2\pi i j k}{p}}\right)_{k=0}^{p-1},\qquad (a_j)_{j=0}^{p-1}\in\mathcal{A}^p.$$

Noncommutative Tao uncertainty principle. For every $x\in\mathcal{A}^p\setminus\{0\}$,

$$\|x\|_0+\|\widehat{x}\|_0\geq p+1.$$

This conjecture seeks to extend Tao's uncertainty principle from complex vectors in prime dimensions to vectors over unital C*-algebras with invariant basis number. Its status is not determined by the supplied context.