Sign Patterns in a Two Colored Partition Companion series

We study two closely related questions arising from the recent work of Andrews and El Bachraoui on the two-color partition series $$S_1(q)=\sum_{n\ge0}s_1(n)q^n=\sum_{a\ge0}q^a(-q^{a+1};q)_\infty^2$$ and its odd companion, denoted by $T_o(q)$. First, for the eta-normalized companion $$C(q)=(q;q)_\infty T_o(q)=\sum_{n\ge0}c(n)q^n,$$ we prove a strong form of the Andrews--El Bachraoui sign conjecture that $\limsup c(n)=+\infty$ and $\liminf c(n)=-\infty$. Second, we construct an involution using the Franklin-type involution of Chen and Liu to combinatorially explain Andrews--El Bachraoui congruence for $s_1(n)$ modulo 4.