Consider in the finite field $\mathbb{F}_{q^3}$ the functions norm and trace $\mathrm{N}_{\mathbb{F}_{q^3}|\mathbb{F}_{q}},\mathrm{Tr}_{\mathbb{F}_{q^3}|\mathbb{F}_{q}}\,:\mathbb{F}_{q^3}\rightarrow\mathbb{F}_{q}$ and let $\lambda\in\mathbb{F}_{q^3}$. How many solutions can have the polynomial $ \mathrm{Tr}_{\mathbb{F}_{q^3}|\mathbb{F}_{q}}(\lambda x^3)=\mathrm{N}_{\mathbb{F}_{q^3}|\mathbb{F}_{q}}(x)$?
Note: from computer tests the only possible case seem to be the trivial case $x=0$.