Asymptotic density conjecture for multidimensional boustrophedons

For $n\geq 1$ and w\in\left\\{+,-\right\\}^n, let $\mathcal{P}_w$ be the set under consideration and let $\mathcal{R}_w^{(\alpha)}$, $1\leq\alpha\leq 6$, be its six disjoint classes, so that $\mathcal{P}_w=\bigsqcup_{\alpha=1}^6\mathcal{R}_w^{(\alpha)}$. In the case $w=+^n$, define the density

p_n^{(\alpha)}=\frac{\\#\mathcal{R}_{+^n}^{(\alpha)}}{\\#\mathcal{P}_{+^n}}.

Asymptotic density conjecture. For every $1\leq\alpha\leq 6$, the limit $p_{\infty}^{(\alpha)}=\lim_{n\rightarrow\infty}p_n^{(\alpha)}$ exists and satisfies

$$p_{\infty}^{(1)}=\frac{1}{\pi},$$ $$p_{\infty}^{(2)}=p_{\infty}^{(5)}=\frac{1}{2}-\frac{1}{\pi},$$ $$p_{\infty}^{(3)}=p_{\infty}^{(4)}=\frac{2}{\pi}-\frac{1}{2},$$ $$p_{\infty}^{(6)}=1-\frac{3}{\pi}.$$

These densities describe the limiting proportions of the six classes inside the Euler-number-enumerated set $\mathcal{P}_{+^n}$; the source presents these limits as conjectural, and no resolution is supplied here.