Vishik-type isotropy gap conjecture for quasilinear quadratic forms

Let $\phi$ be an anisotropic quasilinear quadratic form over $F$ of dimension at least $2$. Write

$$\dim\phi-\mathfrak{i}_{1}(\phi)=2^{r_1}-2^{r_2}+\cdots+(-1)^{s-1}2^{r_s},$$

where $r_1>r_2>\cdots>r_{s-1}>r_s+1>1$. For $1\leq l\leq s$, define

$$D_l=\sum_{i=1}^{l-1}(-1)^{i-1}2^{r_i-1}+\epsilon(l)\sum_{j=l}^s(-1)^{j-1}2^{r_j},$$

where $\epsilon(l)=1$ if $l$ is even and $\epsilon(l)=0$ if $l$ is odd. Here $\mathfrak{i}_0(\phi_L)$ is the isotropy index after extending scalars to a field extension $L/F$.

Vishik-type isotropy gap conjecture. For every field extension $L/F$, either

$$\mathfrak{i}_0(\phi_L)\geq D_l+\mathfrak{i}_1(\phi)$$

or

$$\mathfrak{i}_0(\phi_L)\leq D_l.$$

The statement is expected to extend a theorem of Vishik from nonsingular quadratic forms to the quasilinear characteristic-two setting. The source presents it as an expectation and gives no resolution.