The other inclusion is trivial.

As a consequence of the above result, we get the following corollary.

Denote by $r'$ the dual exponent $1/r+1/r'=1$ and assume that the right convolution operator $RC_K$ is bounded on $ L_{r,m}(G) $. The sesqui-linear pairing on $\operatorname{Co}(L_{r,m}) \times \operatorname{Co}(L_{r',m^{-1}})$ given by $$\langle{T},{T'}\rangle_{\operatorname{Co}(L_{r,m})} = \langle{V_eT},{V_eT'}\rangle_{L_2}$$ is such that the linear map $$T' \mapsto \Big(
                   T \mapsto \overline{\langle T, T' \rangle_{\operatorname{Co}(L_{r,m})}}
                 \Big)$$ is an isomorphism of $\operatorname{Co}(L_{r',m^{-1}})$ onto the anti-linear dual of $\operatorname{Co}(L_{r,m})$.

We identify $\operatorname{Co}(L_{r,m})$ with $\mathcal M_{r,m}$ by the extended voice transform $V_e$, so that the pairing becomes $$\langle{f},{g}\rangle_{L_2} = \int_G f(x) \overline{g(x)}\, dx,\qquad f\in \mathcal
M_{r,m},\, g\in \mathcal M_{r',m^{-1}}.$$ Since $g\in L_{r',m^{-1}}(G) $, clearly $f\mapsto \overline{\langle{f},{g}\rangle_{L_2}}$ is a continuous anti-linear map, which we denote by $\Gamma_g$, on $\mathcal
M_{r,m}$ whose norm is $$\begin{aligned}
  \lVert{\Gamma_g}\rVert &=  \sup\left\{ \lvert{\langle{f},{g}\rangle_{L_2}}\rvert \mid f\in \mathcal
      M_{r,m},\,\lVert{f}\rVert_{L_{r,m}} \leq 1 \right\} \\
      &\leq \sup\left\{\lvert{\langle{h},{g}\rangle_{L_2}}\rvert \mid h\in  L_{r,m}(G) ,\,\lVert{h}\rVert_{L_{r,m}} \leq 1 \right\}=\lVert{g}\rVert_{L_{r',m^{-1}}}.\end{aligned}$$ Next, since $L_{r,m}(G)$ is the dual of $L_{r',m^{-1}}(G)$, there is $h \in L_{r,m}(G)$ with $\|h\|_{L_{r,m}} \leq 1$ such that $\|g\|_{L_{r',m^{-1}}} = \langle h, g \rangle_{L_2}$. Now, setting $c := \|RC_K\|_{L_{r,m} \to L_{r,m}}$ and $f = c^{-1} \cdot RC_K h$, we have $\|f\|_{L_{r,m}} \leq 1$ and $$\langle f, g \rangle_{L_2}
      