Godbole–Goyt–Herdan–Pudwell recurrence conjecture for ordered partitions with blocks of size 2

An ordered partition of $[2k]$ with $k$ blocks of size $2$ is counted by $op_{[2^k]}(123)$ when it avoids the permutation pattern $123$. Godbole–Goyt–Herdan–Pudwell's recurrence conjecture. For $k\geq0$,

$$op_{[2^{k+2}]}(123)= \frac{329k^3+1215k^2+1426k+528}{2(k+2)(2k+5)(7k+5)}op_{[2^{k+1}]}(123)+\frac{3(k+1)(2k+1)(7k+12)}{(k+2)(2k+5)(7k+5)}op_{[2^k]}(123).$$

This conjecture arose from computer-generated recurrence discovery using Zeilberger's Maple package $FindRec$ and gives a second-order linear recurrence for the enumeration of $123$-avoiding ordered partitions with all blocks of size $2$.