A Verma module need not be projective in its block

By [F2] the maximal-label Verma $\mathrm{\Delta }(m)=M(m)$ is projective in ${\mathcal{O}}_{C_m}$ (and in $\mathcal{O}$). By [F1] the other Verma is simple, $\mathrm{\Delta }(-m-2)=M(-m-2)=L(-m-2)$, and the composition factors of $M(m)$ are $L(-m-2)$ and $L(m)$, each with multiplicity one.

By [F3] the cover $P(-m-2)$ is Verma-filtered and BGG reciprocity gives $(P(-m-2):\mathrm{\Delta }(\mu ))=[\mathrm{\Delta }(\mu ):L(-m-2)]$, which is $1$ for $\mu =-m-2$ and $\mu =m$ by [F1] and $0$ for all other $\mu$ because $L(-m-2)$ lies in the block $C_m$ and other linkage classes contribute no composition factors to it. Hence every Verma flag of $P(-m-2)$ has exactly the two factors $\mathrm{\Delta }(m)$ and $\mathrm{\Delta }(-m-2)$, each once.

In such a flag the bottom factor cannot be $\mathrm{\Delta }(-m-2)$: then $P(-m-2)$ would have $\mathrm{\Delta }(m)=M(m)$ as a quotient, and composing with $M(m)\twoheadrightarrow L(m)$ would exhibit $L(m)$ as a simple quotient, contradicting that $P(-m-2)$ has the unique simple quotient $L(-m-2)$ with $m\mathrm{\ne }-m-2$. So there is a subobject $\mathrm{\Delta }(m)\subseteq P(-m-2)$ with quotient $\mathrm{\Delta }(-m-2)$, giving the short exact sequence $0\to \mathrm{\Delta }(m)\to P(-m-2)\to \mathrm{\Delta }(-m-2)\to 0$; it is nonsplit because projective covers are indecomposable by [F3].

The Verma $\mathrm{\Delta }(-m-2)$ is not projective in the block $C_m$: if it were, the epimorphism $P(-m-2)\twoheadrightarrow \mathrm{\Delta }(-m-2)$ would split, contradicting the nonsplitness of step 2.1. Since $\mathrm{\Delta }(m)$ is projective by step 1.1, projectivity indeed depends on the position of the highest weight in the linkage poset, refuting the statement.