Determinantal equations conjecture for hidden Markov model varieties

Let $\Sigma$ be the alphabet, let $\mathbf{g}_{n,d}$ denote the parameterization of the relevant hidden Markov model, and let $p=(p(v))_{v\in\Sigma^n}$ be a string function. For words $v_i,w_j$ with $|w_jv_i|\le n$, form the matrix whose $(i,j)$-entry is $p(w_jv_i)$. Determinantal equations conjecture. Let $n\ge 2d-1$. Then

$$(p(v))_{v\in\Sigma^n}\in\overline{\operatorname{image}(\mathbf{g}_{n,d})}$$

if and only if

$$\text{\rm det~}[p(w_jv_i)]_{1\le i,j\le d+1}=0$$

for all choices of words $v_1,\ldots,v_{d+1},w_1,\ldots,w_{d+1}$ such that $|w_jv_i|\le n$. This gives determinantal equations for the Zariski closure of the hidden Markov model parameterization, addressing the problem of describing that variety by its invariants.