Proper intermediate normal extension of E with admissible weakening

Let $\mathbf{E}$ and $\mathbf{KM}$ be the calculi considered in the paper. A normal extension of $\mathbf{E}$ is an extension satisfying the normality conditions used for these calculi, and the weakening rule is the inference rule $\square\alpha/\alpha$.

Intermediate-extension conjecture. There is a proper normal extension $\mathbf{L}$ of $\mathbf{E}$ such that

$$\mathbf{E}\subsetneq\mathbf{L}\subsetneq\mathbf{KM},$$

and the weakening rule is admissible in $\mathbf{L}$.

The claim asks for a normal logic strictly between $\mathbf{E}$ and $\mathbf{KM}$ retaining admissibility of weakening. The supplied context establishes that there is a continuum of normal extensions of $\mathbf{E}$ closed under weakening, but does not establish the existence of one properly contained in $\mathbf{KM}$; its status therefore remains open here.