Khavinson's gradient-direction conjecture for bounded harmonic functions

Let $h^{\infty}$ denote the space of bounded harmonic functions on the unit ball $\mathbb{B}^n:=\{x\in\mathbb{R}^n:|x|<1\}$. For $x\in\mathbb{B}^n$ and $l\in\mathbb{S}^{n-1}$, let $C(x,l)$ be the best constant such that

$$|\langle\nabla u(x),l\rangle|\leq C(x,l)\sup_{|y|<1}|u(y)|$$

for every $u\in h^{\infty}$, and let $C(x)$ be the best constant such that

$$|\nabla u(x)|\leq C(x)\sup_{|y|<1}|u(y)|.$$

For $x\ne 0$, define the radial unit vector $n_x=\frac{x}{|x|}$. Khavinson's conjecture. For every $x\in\mathbb{B}^n\setminus\{0\}$,

$$C(x)=C(x,n_x).$$

Khavinson's conjecture asserts that the direction of the radial derivative gives the sharp constant for the modulus of the gradient of a bounded harmonic function in the unit ball. The paper presents a proof of the conjecture, so the conjecture is resolved.