= Solution
<Noether's theorem> associates a continuous one-parameter differentiable symmetry of the <action> with an on-shell conserved current. If a first-derivative <Lagrangian density> varies by $\delta\mathcal L=\partial_\mu B^\mu$, the current is $J^\mu=\mathcal L_{\partial_\mu\phi}\delta\phi-B^\mu$. The <integral> of $J^0$ is conserved when the spatial boundary flux vanishes. This connects symmetries to <conservation laws>; time translation, spatial translation and internal <phase> symmetry furnish familiar examples.
For the strict fixed-coordinate invariance in the question, $B^\mu=0$. In one space dimension, <integration by parts> at the <density> level gives
$$
\delta\mathcal L=
\left[\mathcal L_\phi-\partial_t\mathcal L_{\dot\phi}-\partial_x\mathcal L_{\phi'}\right]\delta\phi
+\partial_t(\mathcal L_{\dot\phi}\delta\phi)
+\partial_x(\mathcal L_{\phi'}\delta\phi).
$$
The bracket vanishes by the field equation. Thus $\delta\mathcal L=0$ yields
$$
\boxed{\partial_t\rho+\partial_xJ_x=0,\qquad
\rho=\mathcal L_{\dot\phi}\delta\phi,\qquad
J_x=\mathcal L_{\phi'}\delta\phi.}
$$
In four <spacetime> dimensions the same identity gives $\partial_\mu J^\mu=0$ with $J^\mu=\mathcal L_{\partial_\mu\phi}\delta\phi$. In particular, for the usual scalar kinetic term the spatial current has the minus sign inherited from $\partial^i=-\partial_i$.
For the quartic <density> in $n$ dimensions,
$$
\mathcal L_\phi=-4\lambda\phi^3,\qquad
\mathcal L_{\partial_\mu\phi}=\partial^\mu\phi,
\qquad\boxed{\Box\phi+4\lambda\phi^3=0.}
$$
To evaluate the displayed dilation current, abbreviate $S=(\partial_\mu\phi)(\partial^\mu\phi)$ and $D=x^\nu\partial_\nu$. Product differentiation gives
$$
\begin{aligned}
\partial_\mu J^\mu
&=2S+(D\partial_\mu\phi)\partial^\mu\phi
+(\phi+D\phi)\Box\phi-n\mathcal L-D\mathcal L\\
&=2S-4\lambda\phi^4-n\mathcal L
+(\phi+D\phi)(\Box\phi+4\lambda\phi^3)\\
&=(4-n)\mathcal L+(\phi+D\phi)(\Box\phi+4\lambda\phi^3).
\end{aligned}
$$
Here $D(S/2)=(D\partial_\mu\phi)\partial^\mu\phi$ makes the differentiated kinetic terms cancel, and $D(\lambda\phi^4)=4\lambda\phi^3D\phi$. <On shell>, the <on-shell divergence of a quartic scalar dilation current> reduces to
$$
\boxed{\partial_\mu J^\mu=(4-n)\mathcal L.}
$$
Hence this current is conserved for generic solutions in \b[four <spacetime> dimensions]. Special field configurations with vanishing <density> do not alter the generic dimensional statement.
The subtraction $-x^\mu\mathcal L$ has a symmetry explanation: for $\delta\phi=\phi+D\phi$, direct variation gives $\delta\mathcal L=D\mathcal L+4\mathcal L=\partial_\mu(x^\mu\mathcal L)+(4-n)\mathcal L$. At $n=4$ the <action> is invariant up to a <boundary term>, so $B^\mu=x^\mu\mathcal L$. Strict invariance of the <density> is not required. This is a classical result; no claim about quantum <scale invariance> is needed.
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