Partial flag Toeplitz limit conjecture

Let $d4c3=(n_1,d4c5,n_k)$ be a finite strongly increasing sequence of positive integers, let $n_0=0$, and let $P_{d4c3}^{(n)}$ be the parabolic subgroup associated with the partial flag variety $SL_n/P_{d4c3}^{(n)}$. Consider a weakly decreasing sequence of $n_k$ non-negative real numbers $(d4c3_i)_{i=1}^{n_k}$ such that $d4c3_{n_i}>d4c3_{n_i+1}$ and $d4c3_j=d4c3_{j+1}$ otherwise. Let $u^{d4c4}d4d9\text{in}d4d9\operatorname{Toep}_{d4c4}(d4d9\mathbb R_{d4d9\ge 0})$ have generating function

$$p_{u^{d4c4}}(x)=\frac{1}{\prod_{j=1}^{n_k}(1-d4c3_jx)}=\frac{1}{\prod_{i=1}^{k}(1-d4c3_{n_i}x)^{n_i-n_{i-1}}}.$$

Partial flag Toeplitz limit conjecture. There exists a sequence $u^{(n)}d4d9\text{in}X_{P_{d4c3}^{(n)}}(\mathbb R_{>0})$ converging to $u^{d4c4}$; conversely, every infinite totally nonnegative Toeplitz matrix with a generating function of this form arises as such a limit. This proposes a precise description of limits of positive partial-flag varieties with fixed flag dimensions; the source provides no resolution evidence.