Injectivity conjecture for convolution with the unit-ball indicator

Let $n\ge 2$. For $1 \le p \le \frac{2n}{n-1}$, consider the convolution operator

$$L^p(\mathbb{R}^n)\to L^p(\mathbb{R}^n),\qquad u\mapsto u\star 1_{B_1},$$

where $B_1$ is the unit ball in $\mathbb{R}^n$. Injectivity conjecture. The convolution operator is one-to-one for all $1 \le p \le \frac{2n}{n-1}$. This is the proposed sharp range of exponents for injectivity of convolution with the unit-ball indicator, motivated by the corresponding kernel question for spherical and elliptic operators; the source does not establish the endpoint range.