Andrews' growth conjecture for the coefficients of $v_1(q)$

Let

$$v_1(q):=\sum_{n\ge 0}\frac{q^{n(n+1)/2}}{(-q^2;q^2)_n}=\sum_{n\ge 0}V_1(n)q^n.$$

Andrews' Conjecture 3. The coefficients satisfy

$$|V_1(n)|\to\infty\quad\text{as }n\to\infty.$$

This is one of Andrews' conjectures on the sign and magnitude of the coefficients of the $q$-series $v_1(q)$; the supplied paper states that it proves Andrews' Conjectures 5 and 6, but gives no resolution status for this conjecture.