Asymptotics conjecture for quarter-plane spiral walks

Let $P_m^{{\mathrel{\tikz{\draw (0ex,1ex) -- (0ex,0ex) -- (1ex,0ex) (-0.2ex,0ex)}}}}$ be the number of quarter-plane spiral walks of length $m$. Let $P_{m,\mathbf{x}}^{{\mathrel{\tikz{\draw (0ex,1ex) -- (0ex,0ex) -- (1ex,0ex) (-0.2ex,0ex)}}}}$, $P_{m,\mathbf{y}}^{{\mathrel{\tikz{\draw (0ex,1ex) -- (0ex,0ex) -- (1ex,0ex) (-0.2ex,0ex)}}}}$ and $P_{m,\mathbf{o}}^{{\mathrel{\tikz{\draw (0ex,1ex) -- (0ex,0ex) -- (1ex,0ex) (-0.2ex,0ex)}}}}$ count such walks ending on the $\mathbf{x}$-axis, on the $\mathbf{y}$-axis and at the origin, respectively. Spiral-walk asymptotics conjecture. As $m\to\infty$,

$$P_m^{{\mathrel{\tikz{\draw (0ex,1ex) -- (0ex,0ex) -- (1ex,0ex) (-0.2ex,0ex)}}}}=\frac{8}{\pi}\times\frac{1}{m}\times2^m\times\left(1-\frac{3}{2m}+\mathrm{O}\left(\frac{1}{m^2}\right)\right).$$

Moreover,

$$P_{m,\mathbf{x}}^{{\mathrel{\tikz{\draw (0ex,1ex) -- (0ex,0ex) -- (1ex,0ex) (-0.2ex,0ex)}}}}=\frac{16}{\pi}\times\frac{1}{m^2}\times2^m\times\left(1-\frac{4-(-1)^m}{2m}+\mathrm{O}\left(\frac{1}{m^2}\right)\right),$$ $$P_{m,\mathbf{y}}^{{\mathrel{\tikz{\draw (0ex,1ex) -- (0ex,0ex) -- (1ex,0ex) (-0.2ex,0ex)}}}}=\frac{16}{\pi}\times\frac{1}{m^2}\times2^m\times\left(1-\frac{8-(-1)^m}{2m}+\mathrm{O}\left(\frac{1}{m^2}\right)\right),$$

and

$$P_{m,\mathbf{o}}^{{\mathrel{\tikz{\draw (0ex,1ex) -- (0ex,0ex) -- (1ex,0ex) (-0.2ex,0ex)}}}}=\begin{cases}\displaystyle\frac{64}{\pi}\times\frac{1}{m^3}\times2^m\times\left(1-\frac{15}{2m}+\mathrm{O}\left(\frac{1}{m^2}\right)\right),&m\text{ even},\\[4pt]0,&m\text{ odd}. \end{cases}$$

These asymptotics were obtained from elementary series analysis because rigorous asymptotic derivation was unavailable in the source; the conjecture remains open in the supplied material.