The frame-completeness conjecture for the class of all interpretability frames

Let $\mathfrak F$ be a class of ${\textup{\textbf{IL}}}$-frames, and define

$${\textup{\textbf{IL}}}[\mathfrak F]=\{A\mid \forall F\in\mathfrak F\ \forall^{\sf valuation}V\ \langle F,V\rangle\models A\}.$$

Let $\mathfrak{All}$ be the class of frames in which every $S$-relation implication that follows both from the ${\textup{\textbf{IL}}{\sf{M}}}$ and ${\textup{\textbf{IL}}{\sf{P}}}$ frame conditions is present. The frame-completeness conjecture.

$${\textup{\textbf{IL}}({\rm All})}={\textup{\textbf{IL}}}[\mathfrak{All}].$$

This conjecture asks whether the core interpretability logic is exactly the logic determined by the class $\mathfrak{All}$; the supplied text presents it as a new conjecture and gives no resolution.