Four-arm IIC winding-number variance and central-limit conjecture

Let $\mathcal{A}^{\eta}_4(\eta,1)$ be the four-arm event at mesh size $\eta$, let $\nu_4^{\eta}$ be the four-arm IIC measure, and let $\tilde{\theta}_{\eta}$ be the winding number of the selected arm. Four-arm winding conjecture. Under $P[\cdot\mid\mathcal{A}^{\eta}_4(\eta,1)]$ and $\nu_4^{\eta}$, as $\eta\to0$, both

$$\operatorname{Var}[\tilde{\theta}_{\eta}]=\left(\frac{3}{8}+o(1)\right)\log\left(\frac{1}{\eta}\right)$$

and

$$\frac{\tilde{\theta}_{\eta}}{\sqrt{\frac{3}{8}\log\left(\frac{1}{\eta}\right)}}\mathrel{\to_d}N(0,1).$$

This is presented as the four-arm analogue of established two-arm winding results and as a refinement of an earlier prediction; the source does not report a proof, so the variance asymptotic and the central limit theorem remain conjectural.