Baruah and Sarma's sign conjecture for coefficients of Rogers–Ramanujan products

Let

$$\frac{1}{R^{5}(q)}=\sum_{n=0}^{\infty}A(n)q^{n},\qquad R^{5}(q)=\sum_{n=0}^{\infty}B(n)q^{n},\qquad \frac{R(q^{5})}{R^{5}(q)}=\sum_{n=0}^{\infty}D(n)q^{n},$$

where

$$R(q)=\frac{(q;q^{5})_{\infty}(q^{4};q^{5})_{\infty}}{(q^{2};q^{5})_{\infty}(q^{3};q^{5})_{\infty}}.$$

Baruah–Sarma sign conjecture. For all integers $n\geq 0$,

$$A(5n)<0\quad(n\neq 0),\qquad B(5n)<0\quad(n\neq 0),\qquad D(5n+1)>0.$$

The cases $A(0)$ and $B(0)$ are excluded because $A(0)=B(0)=1>0$. The conjecture concerns the sign patterns of coefficients of higher-order Rogers–Ramanujan products; the surrounding results establish related eventual sign patterns, while these three inequalities were presented as numerical conjectures.