The enumeration conjecture for Type B non-decreasing parking functions

The Type B non-decreasing parking functions are the elements of the submonoid of $\operatorname{NDPF}_{2N}$ generated by $\mu_i=\pi_i\pi_{2N-i}$ for $1\leq i\leq N$, with $\mu_N=\pi_N$. Let $C_j$ denote the $j$th Catalan number. The BNDPF enumeration conjecture. The cardinality of this monoid is

$$|\operatorname{BNDPF}_N|=\sum_{j=0}^N\binom{N}{j}^2C_j.$$

This formula agrees with all values computed through $N=9$ and with the sequence recorded as OEIS A086618, but a proof of the general enumeration was not known in the source.