Generating-function conjecture for greedy (m,r)-Tamari intervals

For integers $m\geq 1$ and $1\leq r\leq m-1$, let $d53c^{(m,r)}_n$ be the set of greedy $(m,r)$-Tamari intervals of size $n$, and let

$$F_{d53c^{(m,r)}}(t,x)=d53c_{d53c^{(m,r)}}(t,x;p_1,p_2,\ldots)$$

be their generating function with the weights defined in the source. Let $\Omega$ denote the operator used in the source, acting on the generating functions as indicated by the displayed identities. Generating-function conjecture for greedy $(m,r)$-Tamari intervals. For $1\leq r\leq m-1$,

$$F_{d53c^{(m,r)}}(t,x)=\bigl(F_{d53c^{(m,0)}}(t,x)+\Omega\bigr)^{r+1}(1).$$

For $r=0$,

$$F_{d53c^{(m,0)}}(t,x)=1+xt\bigl(F_{d53c^{(m,0)}}(t,x)+\Omega\bigr)^{m+1}(1).$$

Equivalently, for $0\leq r\leq m-2$,

$$F_{d53c^{(m,r+1)}}(t,x)=\bigl(F_{d53c^{(m,0)}}(t,x)+\Omega\bigr)F_{d53c^{(m,r)}}(t,x),$$

and

$$F_{d53c^{(m,0)}}(t,x)=1+xt\bigl(F_{d53c^{(m,0)}}(t,x)+\Omega\bigr)F_{d53c^{(m,m-1)}}(t,x).$$

These identities are conjectured from numerical computation and describe the generating functions of the greedy $(m,r)$-Tamari intervals.