Enumeration conjecture for maximal chains without plus-full-sets in Tamari lattices

Let $\mathcal{T}_n$ be the Tamari lattice, and let $\mathcal{N}_i(n)$ denote the set of maximal chains in $\mathcal{T}_n$ of length $n+i$ with no plus-full-sets. Write $\#\mathcal{N}_i(n)$ for its cardinality.

Enumeration conjecture. For all $i\geq-1$,

$$\#\mathcal{N}_{i}(2i+3)=\prod_{j=1}^{i+1}\binom{3j-1}{2},$$

and for all $i\geq0$,

$$\#\mathcal{N}_{i}(2i+2)=\frac{i}{5}\prod_{j=1}^{i+1}\binom{3j-1}{2}.$$

These formulas are based on numerical evidence and give explicit enumerations for the boundary cases used to study maximal chains in Tamari lattices. The source provides no resolution, so the conjecture remains open.