Positivity conjecture for the distribution in the Bôcher-type characterization

Let $u$ and $v$ be as in Theorem, with $v$ the distribution associated to $u$ and supported on the singular set $\Gamma$. For a test function $\phi\in C_c^\infty(B_2)$, write $\tilde{\phi}=\phi|_\Gamma$ for its restriction to $\Gamma$. Positivity conjecture. The distribution $v$ is positive: there exists a positive Radon measure $\mu$ on $\Gamma$ such that

$$\langle-\Delta u,\phi\rangle=\int_\Gamma\tilde{\phi}\,d\mu.$$

This would improve the regularity of the distribution appearing in the Bôcher-type characterization, replacing a general distributional contribution by a positive measure on the singular set. The conjecture is motivated by the positivity of the harmonic function $u$ and the possibility that this positivity annihilates tangential derivatives, but the source provides no resolution.