Classification conjecture for four-dimensional normed spaces of type IV

Let $K^{\lor}$ be the spherical completion of the underlying valued field $K$, let $V_K$ denote the relevant multiplicative value subgroup, and let $\sim$ denote the equivalence relation on $K^{\lor}\setminus K$ used in the classification. For parameters $x,x_1,y,y_1,z,z_1\in K^{\lor}\setminus K$ and $t,t_1,s,s_1\in\mathbb{R}_{>0}$, define

$$(E,\|\cdot\|):=([(1,0),(0,1),(x,0),(y,z)],|\cdot|_t\times|\cdot|_s),$$ $$(E_1,\|\cdot\|):=([(1,0),(0,1),(x_1,0),(y_1,z_1)],|\cdot|_{t_1}\times|\cdot|_{s_1}).$$

Suppose that both $E$ and $E_1$ are of type $\mathrm{IV}_4$. Classification conjecture. The spaces $E$ and $E_1$ are isomorphic if and only if:

$$\frac{t}{t_1},\frac{s}{s_1}\in V_K;$$ $$x\sim x_1,\qquad z\sim z_1;$$ $$([1,x,y],|\cdot|)\cong([1,x_1,y_1],|\cdot|).$$

This would provide a classification of four-dimensional normed spaces of type $\mathrm{IV}_4$, a structure theorem for which is currently unknown; the source presents the assertion as an expected conjecture, and no resolution is supplied.