The generating-function conjecture for capacity-two multiplex juggling cards

Let $a_b^{(2)}$ be the number of cards for multiplex juggling with $b$ balls, where each track has capacity at most $2$.

Generating-function conjecture.

$$\sum_{b\ge 0}a_b^{(2)}x^b = \frac{1-x+x^2+x^3}{(1-x-x^2)^3}.$$

The sequence was observed to satisfy this identity through $b=25$; the surrounding discussion notes that analogous counting questions for multiplex juggling with higher capacity have less clear behavior and that these counts had not appeared in the OEIS.