It is not true. Take $\mathfrak{sl}_3$, with simple reflections $s,t$, such that $s$ is singular. Put $x=e$, $w= st$; both are in $W^\Sigma$.
I claim that $Ext_\mathcal{O}^i(M(\mu),L(st \cdot \mu)) = 0$ for all $i \geq 0$.
Denote by $\lambda$ a regular, integral antidominant weight. By [2, Corollary 1.3.3.] we have
$$ \dim Ext_\mathcal{O}^i(M(\mu),L(st \cdot \mu)) = \begin{cases} \dim Hom(M(\lambda),L(st \cdot \lambda))=0 & \colon i=0 \\ \dim Ext_\mathcal{O}^i(M(\lambda),L(st \cdot \lambda)) - \dim Ext_\mathcal{O}^{i-1}(M(s \cdot \lambda),L(st \cdot \lambda)) &:i>0 \end{cases}$$.
We know that $P_{e,st} = P_{s,st} = 1$. From KL polynomial in terms of extension group in the regular case [1, Theorem 8.11.], it follows that the above always cancels. (For $i=2$ we have $1-1=0$.)
[1] Humphreys: Representations of semisimple Lie algebras in the BGG category O. Graduate Studies in Mathematics, 94. American Mathematical Society, Providence, RI, 2008.
[2] Irving: Singular blocks of the category O, Math Z (1990) 204: 209.