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

Let

$$\rho(q)=\sum_{n\geq 0}r(n)q^n$$

be defined by

$$\rho(q)=\sum_{m\geq 0}\frac{q^{2m(m+1)}}{(1+q+q^2)(1+q^3+q^6)\cdots(1+q^{2m+1}+q^{4m+2})}.$$

The finite sign law. For every $n\geq 0$,

$$r(3n)>0,$$ $$r(3n+1)\leq 0,$$

and

$$r(3n+2)\leq 0.$$

Moreover, the only zeros in the last two families are

$$r(2)=r(4)=r(8)=r(11)=r(20)=0.$$

The asymptotic version is proved in the paper, but determining the exact finite cutoff remains separate; the stated finite law is supported experimentally and is not established by the supplied text.