OPE coefficients of descendants in the presence of null vectors
Consider the OPEs of a 2D CFT, with nondegenerate primary fields labeled by some collection $\mathtt{Prim}$ of indices. The data specifying the OPE can be taken as a collection $${C_{p,q}^{r;P}}{p,q,r\in \mathtt{Prim},P \in \mathrm{Part}}\subset \mathbb{C},$$ where $\mathrm{Part}$ is the collection of all multisets of positive integers (including the empty multiset). Hence, an element $P\in\operatorname{Part}$ can be uniquely identified with a partition of some nonnegative number into positive parts and vice versa. (To each such partition we can associate an element $L{P}=L_{-n_1}\cdots L_{-n_K}$, where $n_1\geq\cdots\geq n_K\geq 1$ are the elements of $P$, with multiplicity.) The complex number $C^{r;P}_{p,q}$ is then the coefficient of $L_P$ applied to the $r$th primary field in the OPE between the $p$th and $q$th primary fields.
Given this setup, we can solve for $C^{r;P}{p,q}$ in terms of $C^r{p,q}=C^{r;\varnothing}{p,q}$ using conformal invariance. See [1], Section 6.6.3. However, this algorithm seems to use non-degeneracy quite centrally, as solving for the coefficients $C^{r;P}{p,q}$ at level $N$ requires inverting the Kac-Gram matrix at level $N$. Concretely, in order to solve Eq. 6.180 in [1], one must invert the matrix on the left-hand side, which is essentially the Kac-Gram matrix. 945
Clearly, this fails when discussing the analogous problem in the presence of degenerate fields, e.g. when working with minimal models, since the Kac-Gram matrix is non-invertible. (The denominators in Eq. 6.181 can vanish, for instance.) This is discussed briefly in [1], Section 8.A, where the connection is made to the fusion rules of minimal models. However, I do not think that book contains a proof of something like the following, analogous to the result that applies when the Kac-Gram matrix is invertible at every level:
Letting ${C^{r;P}{p,q}}{p,q,r,P}$ denote a set of indended OPE coefficients for a theory with the primary field content specified by the minimal models, each $C^{r;P}{p,q}$ is uniquely determined by the collection ${C^{r;\varnothing}{p,q}}{p,q,r}$ (up to the coefficients of null fields), and conversely given ${C^{r;\varnothing}{p,q}}{p,q,r} \subset \mathbb{R}$ (which may be subject to some specified finite set of constraints) there exists some collection ${C^{r;P}{p,q}}_{p,q,r,P}$ of numbers such that the corresponding OPE is conformally invariant in the sense of Section 6.6 of [1].
Since this seems important to the construction of minimal models via the bootstrap, I assume it is true, but it does not appear straightforward to extract a proof from the content of [1]. Is there somewhere I can find a proof of this result? Requiring that the right-hand side of the $N$th level analogue of Eq. 6.180 is in the range of the matrix on the left-hand side for all $N$ naively imposes an infinite number of constraints on the conformal dimensions $h_p,h_q,h_r$ associated to $p,q,r$ when the $r$th primary field is degenerate, so from this perspective the proposition above would seem quite miraculous.
If one defines the term "fusion rules" to refer to the triples of $h_r,h_p,h_q$ such that the proposition holds, then the question becomes to show that the restrictions on $h_r,h_p,h_q$ described in Chapter 7 pf [1] are actually the fusion rules. (E.g., why does the discussion in Section 7.3 suffice? 945