This is question is about whether or not certain modules for an affine Lie algebra are generated by their singular vectors. I begin with some background.
Backround on affine Lie algebras. Let $\mathfrak{g}$ be a simple Lie algebra over $\mathbf{C}$. Fix a level $\kappa\in (S^{2}\mathfrak{g}^{*})^{\mathfrak{g}}\cong H^{2}(\mathfrak{g}((t));\mathbf{C}).$ We identify the vector space of levels with $\mathbf{C}$ via the choice of basic form, $\kappa_{b}$, which is fixed by $\kappa_{b}(\theta^{\vee},\theta^{\vee})=2.$ This determines the affine Lie algebra at level $\kappa$, $\hat{\mathfrak{g}}_{\kappa}$ as the corresponding central extension. This is graded so that $x\otimes t^{n}$ has weight $-n$. Consider the category of graded modules for $\hat{\mathfrak{g}}_{\kappa}$ with weights in $\mathbf{Z}_{\geq 0}$ and all weight spaces finite dimensional. Denote this category $\mathcal{C}_{\kappa}$. If $M\in\mathcal{C}_{\kappa}$ we call a vector $m\in M$ singular if $(x\otimes t^{n})(m)=0$ for all $x\in\mathfrak{g}$ and $n> 0$.
Question. If $M\in\mathcal{C}_{\kappa}$, is $M$ necessarily generated by its singular vectors as a $\hat{\mathfrak{g}}_{\kappa}$ module?
Remark. A trick, presumably well known by people in the field, with the $0$-mode of the Sugawara vector and eigenvalues of the Casimir shows that this is true if $\kappa\notin\kappa_{\operatorname{crit}}+\mathbf{Q}_{>0}.$ In fact in this case $\mathcal{C}_{\kappa}$ seems just to be $\operatorname{Vect}^{\operatorname{fin}}$, so really it is the so-called positive case of $\kappa\in\kappa_{\operatorname{crit}}+\mathbf{Q}_{>0}$ which is the most interesting.
Edited to remove a a-priori stronger form of the question which is in fact obviously equivalent to it.