Sign law for Ramanujan's third order mock theta function $\rho (q)$

We study the coefficients of Ramanujan’s third order mock theta function $$\rho (q)=\sum _{m\ge 0}\frac{q^{2m(m+1)}}{(1+q+q^2)(1+q^3+q^6)\cdots (1+q^{2m+1}+q^{4m+2})}=\sum _{n\ge 0}r(n)q^n\mathrm{.}$$ Numerical evidence suggests the striking sign pattern $$r(3n)>0,r(3n+1)\le 0,r(3n+2)\le 0.$$ We prove an asymptotic form of this phenomenon. More precisely, using Watson’s relation between $\rho (q)$ and $\omega (q)$, together with a Rademacher-type expansion for the coefficients of $\omega (q)$ and the corresponding expansion for a theta–eta product, we show that $$r(n)\sim {\kappa }_{n3}\frac{2\pi }{(12n+8{)}^{1\mathrm{/}4}}{I}_{1\mathrm{/}2}\text{\hspace{0.17em}⁣}\left(\frac{\pi \sqrt{12n+8}}{18}\right),$$ where $${\kappa }_0=\frac{1}{3}\cos \frac{\pi }{18}>0,{\kappa }_1=-\frac{1}{3}\sin \frac{2\pi }{9}<0,{\kappa }_2=-\frac{1}{3}\sin \frac{\pi }{9}<0.$$ Consequently, $$r(3n)>0,r(3n+1)<0,r(3n+2)<0$$ for all sufficiently large $n$.