The lax- and pseudo-slice characterization of 2-limits

Let $I$ and $\mathcal A$ be $2$-categories, let $F\colon I\to\mathcal A$ be a $2$-functor, and let $L\in\mathcal A$ be equipped with a $2$-natural transformation $\lambda\colon\Delta L\Rightarrow F$. The claim concerns the lax-slice and pseudo-slice $2$-categories of $2$-cones over $F$.

Lax- and pseudo-slice characterization. The pair $(L,\lambda)$ is a $2$-limit of the $2$-functor $F$ precisely when it is a $2$-terminal object in the lax-slice (or pseudo-slice) $2$-category of $2$-cones over $F$.

The paper states that this conjecture is also incorrect: the pseudo-slice may fail to detect enough modifications while introducing too many morphisms, and analogous problems occur for the lax-slice.