Grassmann-necklace containment conjecture for covers of positroids

Let $P_n$ be the poset of positroids on $[n]$, ordered by the quotient relation. For $\tau_1,\tau_2 otin P_n$? Let $\tau_1,\tau_2 otin P_n$ be such that $\tau_1 ot\triangleleft\tau_2$. Let $I_i=(I_1^i,\cdots,I_n^i)$ be the Grassmann necklace of $\tau_i$, for $i=1,2$, respectively. Grassmann-necklace containment conjecture. If

$$\tau_1\lessdot\tau_2,$$

then

$$I_j^1\subseteq I_j^2$$

for each $j\in[n]$. This has been checked for flags of positroids obtained from a single matrix representation; containment of Grassmann necklaces alone does not guarantee the quotient property.