Rob Pratt's generating-function conjecture for the packing number of the double vertex graph of a path

Let $A085680(n)$ denote the packing number of the double vertex graph of the path graph $P_n$, equivalently $\rho(F_2(P_n))$, where $F_2(P_n)$ is the 2-token graph of $P_n$. Rob Pratt's conjecture asserts

$$\sum_{n\geq 0} A085680(n+2)x^n=\frac{1-x+x^2-x^{10}+x^{11}}{(1-x)^2(1-x^5)}.$$

This conjecture gives the ordinary generating function for the sequence whose exact values were known up to $n=50$ at the time it was posed; the supplied source does not indicate whether the conjecture has since been resolved.