Does non hyponormal operator implies non quasinormal operator

Question

I read that every quasinormal operator is hyponormal. I constructed an example which is not hyponormal but it turns out that it was quasinormal! The example is this: Take $\varphi(z)= 2 z^{3}+ \bar{z}^3.$ $T_{\varphi}$ be toeplitz operator with symbol $\varphi$. For $T_{\varphi}$ to be hyponormal .I need $\langle T^* T x, x \rangle \geq \langle T T^* x ,x \rangle. \forall x \in F_{m}^{2} ,\text{which is Fock-Sobolev space.} $ I found out that for m=1, $T_{\varphi}(z^3) =2 z^6 + 24, T_{\varphi}^* (z^3) = z^6 + 48$ and $\langle T^* T z^3, z^3 \rangle = 580 , \langle T T^* z^3 ,z^3 \rangle = 2305.$ Clearly,$ \langle T^* T z^3, z^3 \rangle \leq \langle T T^* z^3 ,z^3 \rangle. $ So not hyponormal, but for quasinormal ,$T_{\varphi} ^2 (z^3) = 4 z^9 +468 z^3$. and $ T_{\varphi }^* T_{\varphi} (z^3) =1704 z^3 +2 z^9 .$ $\langle T_{\varphi} T_{\varphi}^{*} T_{\varphi} z^3,\; z^3 \rangle = 0.$ $ \langle(T_{\varphi^*}T_{\varphi}^2-T_{\varphi}T_{\varphi^*}T_{\varphi})(z^3), \; z^3 \rangle = 0.$ SO, i got that it was quasinormal. I am confused any help will be appreciated. Thanking you in advance.