Scully's refined splitting-pattern conjecture for quadratic forms

Let $F$ be a field of characteristic different from $2$, and let $p$ and $q$ be anisotropic quadratic forms over $F$ of dimension at least $2$. Let $h(q)$ be the height of the splitting pattern of $q$, and let $q_0,q_1,\ldots,q_{h(q)}$ be its successive higher anisotropic kernels. Set $k=\operatorname{dim}(q)-2\mathfrak{i}_W(q_{F(p)})$, and choose $l$ with $0\leq l\leq h(q)$ and $\operatorname{dim}(q_l)=k$. Let $s$ be the unique non-negative integer satisfying

$$2^s<\operatorname{dim}(p)\leq 2^{s+1},$$

and suppose that $k<2^s$. For each $r$ with $0\leq r<l$, Scully's refined conjecture. There exist integers $a_r,b_r\geq1$ and integers $\epsilon_r$ with $-k\leq\epsilon_r\leq k$ such that

$$\operatorname{dim}(q_r)=2^{s+1}a_r+\epsilon_r,$$

and, with at most one exception, either

$$\mathfrak{i}_{r+1}(q)\leq\frac{1}{2}(k+\epsilon_r)$$

or

$$\mathfrak{i}_{r+1}(q)=b_r2^{s+1}+\epsilon_r.$$

The possible exception is when $\operatorname{dim}(q_r)=2^{s+1}-k$; then $r=l-1$ and $\mathfrak{i}_{r+1}(q)=2^s-k$. This conjecture refines the dimension-and-Witt-index conjecture to the full splitting pattern of $q$. The source proves it when $k\leq7$ or when $\operatorname{dim}(q)\leq2^{s+2}+k$, while the general statement was presented as conjectural.