Infinitely many negative terms in two weighted partition sequences

Recently, Andrews and El Bachraoui introduced partition-counting functions a₂(n) and b₂(n), together with their signed analogues a'₂(n) and b'₂(n), and proved that a'₂(n), b'₂(n)>0 for all n≥ 3. For m≥ 2, the latter functions extend naturally to a'ₘ(n) and b'ₘ(n), defined by {align*} ∑n≥ 0 a'_m(n)q^n &= -q/1-q (q^m;q^m)_∞/(q;q^m)_∞ +q/(1-q)^2,\\ ∑n≥ 0 b'_m(n)q^n &= -{q(1-qᵐ⁻¹)}{1-q} {(q²ᵐ;q^m)_∞}{(q;q^m)_∞} +q/1-q ( 1/1-q -{qᵐ⁻¹}{1-qᵐ⁺¹} ). {align*} They asked whether there exists an integer $m>2$ for which a'ₘ(n) is eventually positive, and posed the same question for b'ₘ(n). In this paper, we resolve these two questions.