Monotonicity conjecture for invariants of unconstrained hidden Markov models

Let $\mathbf{f}_{n,l}$ be the unconstrained hidden Markov model with $l$ hidden states and strings of length $n$, and let $d(n,l)$ denote the maximum degree of its invariants. Monotonicity conjecture. For $n\ge 2l$, the maximum degree does not increase as the string length grows:

$$\cdots\le d(n+1,l)\le d(n,l)\le d(n-1,l)\le\cdots\le d(2l,l).$$

The conjecture concerns stabilization of invariant degrees for hidden Markov models. The supplied text notes that an ideal-theoretic analogue of a preceding lifting theorem would imply a related conjecture, but does not establish this degree monotonicity statement, so its resolution is left open here.