Proving WZW primaries are Virasoro primaries

Question

It is stated (for example, in Di Francesco, Mathieu, and Senechal's CFT book, section 15.3.1) that if a field $\phi$ is a "WZW primary", that is, it has OPE

$$J^a(z) \phi(w) \sim -\frac{t^a \phi(w)}{z-w}$$

where $J^a$ are the chiral currents in a WZW theory and $t^a$ are some representation of the generators of the corresponding Lie algebra, then $\phi$ is a Virasoro primary. The "proof" that I can find (see the same section of the same book) only proves that $L_0|\phi \rangle = h |\phi \rangle$, where $L_n$ are Virasoro modes and $h = C/2(k+g)$. ($C$ is the quadratic Casimir $t^a t^a$ of the representation, $k$ is the level of the theory, and $g$ is the dual Coxeter number.) This indeed proves that $\phi$ is a scaling field. However, it does not tell me that I get the correct action of $L_{-1}$ to actually see that $\phi$ is primary, or equivalently that the OPE with the energy-momentum tensor has a term $\partial \phi/(z-w)$. How can I see this fact?

I have tried calculating the OPE of $\phi$ with the Sugawara energy-momentum tensor directly; what I got was

$$T(z) \phi(w) \sim \frac{1}{2(k+g)} \left(\frac{C \phi(w)}{(z-w)^2} - \frac{2 t^a :J^a \phi:(w)}{z-w}\right)$$

where the colons denote normal ordering and repeated indices are summed. The first term gives the correct scaling dimension as expected, but (assuming I did the calculation correctly) I don't know what to do with the normal ordered product in order to get $\partial \phi/(z-w)$. Thanks!

EDIT: I think this is actually false in general. Consider two decoupled CFTs, the WZW and some second theory CFT$_2$ with the same speed of light so that the tensor product theory, with energy-momentum tensor $T = T_{WZW} + T_{CFT_2}$, is conformally invariant. Assuming everything in CFT$_2$ is a singlet under the action of the group defining the WZW theory, pick a WZW primary $\phi$ and any CFT$_2$ Virasoro primary $\psi$. Then it's easy to see from the OPEs that the field $\phi \otimes \psi$ is a WZW primary but not a primary of the WZW theory's Virasoro. (However, if $\phi$ is itself a primary of the WZW's Virasoro, then $\phi \otimes \psi$ is of course a primary of the larger theory.)

A modification of the question: if the WZW theory is not embedded in a larger theory, then are WZW primaries always Virasoro primaries?