Eventual strict increase conjectures for sixfold Thue–Morse representations

Let ${\bf t}=t_0t_1t_2\cdots$ be the Thue–Morse sequence. For $j\geq 2$, define

$$r_j(n):=\left|\{(x_1,\ldots,x_j):n=\sum_{1\leq i\leq j}x_i\text{ and }t_{x_i}=0\text{ for }1\leq i\leq j\}\right|$$

and

$$s_j(n):=\left|\{(x_1,\ldots,x_j):n=\sum_{1\leq i\leq j}x_i\text{ and }t_{x_i}=1\text{ for }1\leq i\leq j\}\right|.$$

Sixfold strict increase conjectures. The sequences $r_6(n)$ and $s_6(n)$ satisfy

$$r_6(n)<r_6(n+1)\qquad\text{for }n\geq 37,$$

and

$$s_6(n)<s_6(n+1)\qquad\text{for }n\geq 5.$$

Numerical experiments suggest that both sequences are eventually strictly increasing; the conjectured thresholds are $37$ for $r_6$ and $5$ for $s_6$.