Indeed, defining $\underset{\_}{{\eta }_{+}}=\frac{1}{\sqrt{t}}{\eta }_{+}$ and $\underset{\_}{{\eta }_{-}}=\sqrt{t}{\eta }_{-}$, we see that the OPEs of $\underset{\_}{{\eta }_{+}}$ and $\underset{\_}{{\eta }_{-}}$ are exactly the same as those of ${\eta }_{+}$ and ${\eta }_{-}$. Moreover, $\varphi (\underset{\_}{{\eta }_{+}})=\underset{\_}{{\eta }_{-}}$ and $\varphi (\underset{\_}{{\eta }_{-}})=\underset{\_}{{\eta }_{+}}$.

From the OPE in Appendix pp2, it follows immediately that $\varphi (W)=sW$ with $s=\pm 1$. We then compute $(W,W)=sc/3$, and use (15) to deduce that $s=-(-1{)}^{h_W+R_W-r_W}=(-1{)}^{R_W}$, where we used $c<0$ and $r_W=0$. The latter follows from the relation $r_W=-r_{\varphi (W)}$. To determine $R_W$, we first note that $R_W<h_W=3$, since the relation $R=h$ holds only for Higgs branch operators, which are absent in this theory. Thus, the possible values are $R_W=0$, 1, 2. We then use the fact that, for a singular term in the OPE, the $R$-charges satisfy $R_{\{AB{\}}_n}\le R_A+R_B-1$ for $n>0$. Applying this to the term $\mathrm{\Lambda }\subset W\times W$, where $\mathrm{\Lambda }$ is the quasiprimary operator built from $T$ with $h_{\mathrm{\Lambda }}=4$ and $R_{\mathrm{\Lambda }}=2$, uniquely fixes $R_W=2$, and hence $s=1$. This also suggests that $W$ arises from the multiplet ${\widehat{\mathcal{C}}}_{1(0,0)}$.

Since $0\le R\le h$, the condition $R_{\mathrm{\Psi }},r_{\mathrm{\Psi }}\in 2\mathbb{Z}+\frac{1}{2}$ implies that $R_{\mathrm{\Psi }}=\frac{1}{2}$ or $\frac{5}{2}$. The possibility $R_{\mathrm{\Psi }}=\frac{1}{2}$ can be excluded, since it would require $\mathrm{\Psi }$ to belong to a multiplet of type ${\mathcal{D}}_{0(0,j_2)}$ or ${\overline{\mathcal{D}}}_{R(j_1,0)}$, and such multiplets occur only in free theories, with $j_i=0$. We, therefore, conclude that $R_{\mathrm{\Psi }}=\frac{5}{2}$. Moreover, the structure of the Macdonald index at order $q^4$ takes the form $q^4(T+T^2-T^2-T^3)$, where the term $q^4{T}^2$ corresponds to the quasiprimary operator $\mathrm{\Lambda }$. The remaining negative contributions must then come from $\mathrm{\Psi }$ and $\tilde{\mathrm{\Psi }}$, which implies $R_{\mathrm{\Psi }}-r_{\mathrm{\Psi }}=2$ and $R_{\mathrm{\Psi }}+r_{\mathrm{\Psi }}=3$. Hence, $R_{\mathrm{\Psi }}=\frac{5}{2}$ and $r_{\mathrm{\Psi }}=\frac{1}{2}$. Examining the multiplet structure of Schur operators, one then finds that $\mathrm{\Psi }\in {\widehat{\mathcal{C}}}_{\frac{3}{2}(0,\frac{1}{2})}$ and $\tilde{\mathrm{\Psi }}\in {\widehat{\mathcal{C}}}_{\frac{3}{2}(\frac{1}{2},0)}$.