The McKinsey axiom is not d-persistent

Let $p$ be a proposition letter, and let $\hspace{.2ex}\text{ \tikz[baseline=-.6ex, rounded corners=.01ex, line width=.12ex] {\draw (-.6ex,-.6ex) rectangle (.6ex,.6ex);}}\kern.2ex$ and $\hspace{.2ex}\text{ \tikz[baseline=-.6ex, rounded corners=.01ex, rotate=45, line width=.12ex] {\draw (-.5ex,-.5ex) rectangle (.5ex,.5ex);}}\kern.2ex$ denote the modal operators. A modal consequence pair is d-persistent when it is preserved under the d-persistence construction considered in the paper. The McKinsey conjecture. The McKinsey axiom

$$\hspace{.2ex}\text{ \tikz[baseline=-.6ex, rounded corners=.01ex, line width=.12ex] {\draw (-.6ex,-.6ex) rectangle (.6ex,.6ex);}}\kern.2ex\hspace{.2ex}\text{ \tikz[baseline=-.6ex, rounded corners=.01ex, rotate=45, line width=.12ex] {\draw (-.5ex,-.5ex) rectangle (.5ex,.5ex);}}\kern.2ex p \mathrel{\trianglelefteqslant} \hspace{.2ex}\text{ \tikz[baseline=-.6ex, rounded corners=.01ex, rotate=45, line width=.12ex] {\draw (-.5ex,-.5ex) rectangle (.5ex,.5ex);}}\kern.2ex\hspace{.2ex}\text{ \tikz[baseline=-.6ex, rounded corners=.01ex, line width=.12ex] {\draw (-.6ex,-.6ex) rectangle (.6ex,.6ex);}}\kern.2ex p$$

is not d-persistent. This conjecture concerns the persistence of modal consequence pairs beyond distributive modal logic; the paper notes that no explicit example of a modal consequence pair failing d-persistence is known and expects a proof inspired by the cited work.