= Solution
For minimization, the <optimization Lagrangian> is
$$
L=3y-z+\lambda(2x-y-z-2)+\mu(x^2+y^2-5),\qquad \lambda\geq0.
$$
Its coefficient of $z$ is $-1-\lambda<0$, so its infimum over unrestricted $z$ is $-\infty$ for every allowable <Lagrange multiplier>. There is no finite global minimizer of the <optimization Lagrangian> to which the <Lagrangian sufficiency theorem> could apply.
The primal <minimization problem> is also unbounded: set $x=0$, $y=\sqrt5$ and $z=t\geq0$. These points are feasible, and $3y-z=3\sqrt5-t\to-\infty$. Hence
$$
\boxed{\inf(3y-z)=-\infty.}
$$
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