Selmer's cubic is a counterexample to the Hasse principle for ternary cubic forms.
We also know that the Hasse principle does not hold for quaternary cubic forms, as
$$ 5x^3 + 12y^3 + 9z^3 + 10t^3 $$
represents zero over every completion of $\mathbb{Q}$, but does not represent zero over $\mathbb{Q}$.
My question is: Is there a known explicit counterexample to the Hasse principle of the form:
$$ x^3 + y^3 + z^3 + nt^3, $$
for some $n \in \mathbb{Z}$, and would a demonstration that the Hasse principle holds for cubic forms of this form not trivially imply the "sums of three cubes conjecture"?
By this I of course mean that $x^3 + y^3 + z^3$ is universal over $\mathbb{Q}_p$ for every $p \neq 3$, and hence represents $-n$ over $\mathbb{Q}_p$ for every $p \neq 3$. The representation of $0$ by $x^3 + y^3 + z^3 + nt^3$ over $\mathbb{Q}_3$ follows from the congruence conditions on $n$ mod 9, and thus the representation of $0$ by $x^3 + y^3 + z^3 + nt^3$ over $\mathbb{Q}$ follows from the Hasse principle.
I am, of course, aware that
$$\{ x^3 + y^3 + z^3 + nt^3 \mid n \in \mathbb{Z} \} $$
is a very artificially constructed class of cubic forms.
Many thanks.